TO  MEMORIAL 


rviiis  Str 


Math.  Dt0t> 


NEW    METHOD 


CORRECTING  LUNAR  DISTANCES, 


AND 


IMPROVED   METHOD 


OF  FINDING  THE 


ERROR  AND  RATE  OF  A  CHRONOMETER 


BY 


EQUAL  ALTITUDES. 


BY  WILLIAM  CHAUVENET,  LL.  D., 

CHANCELLOE  OF  WASHINGTON  UNIVERSITY,  ST.  LOUIS,  LATE  PEOFESSOB  OF  MATHEMATICS  AND  ASTEONOMY 

IN  THE  U.  S.  NAVAL  ACADEMY,  ETC. 


EXTRACTED  FROM  THE  APPENDIX  TO  THE  AMERICAN  EPHEMERIS  FOR  1857,  BY  AUTHORITY  OF  THE 
BUREAU  OF  NAVIGATION,  NAVY  DEPARTMENT,  WASHINGTON. 


NEW  YORK: 

D.   VAN    NOSTRAND,    192    BROADWAY. 
LONDON  :  TBUBNEE  &  CO. 

1868. 


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. 
NEW    METHOD 


OF 


CORRECTING   LUNAR  DISTANCES 


FROM  THE  APPENDIX  TO  THE  AMERICAN  EPHEMERIS  AND  NAUTICAL 
ALMANAC  FOB  1855. 


812165 


•   • 


DIRECTIONS 


FOR  USING 


THE  TABLES  FOR  CORRECTING  LUNAR  DISTANCES. 


THE  object  of  these  Tables  is  to  give  the  true  correction  of  a  lunar  distance  in  all 
cases  when,  with  the  apparent  distance  of  the  moon  from  the  sun,  a  planet,  or  star, 
the  apparent  altitudes  of  the  two  objects  have  also  been  obtained  by  observation. 
They  enable  us  readily  to  take  into  account,  —  1st,  the  parallax  of  the  moon  in  the 
latitude  of  the  observer,  allowing  for  the  spherpidal  figure  of  the  earth ;  2d,  the 
parallax  of  the  sun  or  a  planet ;  3d,  the  true  atmospheric  refraction,  allowing  for  the 
actual  state  of  the  air  as  shown  by  the  barometer  and  thermometer ;  and  4th,  that 
effect  of  refraction  which  gives  the  apparent  discs  of  the  moon  and  sun  an  oval  or 
elliptical  figure. 

The  longitude  deduced  from  a  iunar  observation,  when  no  attention  is  paid  to  the 
spheroidal  figure  of  the  earth,  to  the  barometer  and  thermometer,  and  the  elliptical 
figure  of  the  discs,  may  in  certain  cases  be  in  error  a  whole  degree.  It  is  true  these 
extreme  cases  are  rare  in  practice;  but  cases  are  common  in  which  from  such 
neglect  the  error  in  the  longitude  is  10',  15',  or  20'.  Since  lunars  are  now  chiefly 
valuable  as  checks  upon  the  chronometer,  it  is  absolutely  necessary  to  get  rid  of  such 
errors,  and  to  leave  no  other  inaccuracy  in  the  result  than  that  which  unavoidably 
follows  from  the  observations.  This  is  accomplished  by  means  of  these  Tables,  with 
an  amount  of  labor  very  little  greater  than  that  which  is  required  by  the  inaccurate 
methods  in  common  use. 

THE  OBSERVATION. 

The  record  of  a  complete  observation  embraces,  — 

1.  The  latitude  and  approximate  longitude  of  the  place  of  observation. 

2.  The  approximate  local  time. 

3.  The  time  of  observation  as  shown  by  a  chronometer,  and  the  error  of  the  chro- 
nometer, or  its  difference  from  mean  Greenwich  time. 

4.  The  apparent  distance  of  the  moon's  b  *ight  limb  from  a  star  or  planet,  or  from 
the  nearest  limb  of  the  sun. 

5.  The  apparent  altitude  of  the  moon's  upper  or  lower  limb  above  the  sea  horizon. 

6.  The  apparent  altitude  of  the  star,  planet,  or  lower  limb  of  the  sun  above  the 
sea  horizon. 

5 


CORRECTION   OF 

•*  - 
_ 

7,  The  height  of  the  barometer  and  thermometer. 
' ::  .-••  8.  The  hei-ght  of  the  eye  above  the  level  of  the  sea. 

9.  The  index  correction  of  the  sextant,  if  a  sextant  is  used. 

I  *.?<[  '!J)he i  indfix  corrie*ction  of  the  sextant  may  be  supposed  to  be  previously  determined  , 
but,  since  even  in  the  best  instruments  it  is  not  constant,  its  determination  should  be 
considered  a  necessary  part  of  the  observation ;  and  when  the  greatest  precision  is 
sought,  it  should  be  found  both  before  and  after  the  measurement  of  the  distance,  and 
its  mean  value  taken. 

The  error  of  the  chronometer  above  alluded  to  is  that  which  is  obtained  by  apply- 
ing the  daily  rate  (multiplied  by  the  proper  number  of  days)  to  the  error  found  before 
leaving  port.  The  agreement  or  disagreement  of  the  error  thus  found  with  that 
found  by  the  lunar  observation  will  be  the  test  of  the  good  or  bad  going  of  the 
chronometer. 

PREPARATION  OF  THE  DATA. 

Greenwich  Date.  —  Correct  the  chronometer  time  for  its  error  from  Greenwich 
time  and  deduce  the  Greenwich  Date,  i.  e.  the  Greenwich  day  and  hour  (mean  time), 
reckoning  the  hours  in  succession»from  0  to  24,  beginning  at  noon. 

Nautical  Almanac.  —  With  the  Greenwich  Date  enter  the  Almanac  and  take  out 
the  moon's  semidiameter  and  horizontal  parallax ;  and  if  the  sun  is  observed,  its  semi- 
diameter  and  horizontal  parallax  ;  *  but  if  a  planet  is  observed,  its  horizontal  parallax 
only. 

Apparent  Altitude  of  the  Moon.  —  To  the  altitude  given  by  the  sextant  apply  the 
index  correction  of  the  instrument  and  subtract  the  dip  of  the  horizon,  Table  I.  If 
the  lower  limb  is  observed,  add  the  semidiameter  augmented  by  Table  II. ;  if  the 
upper  limb  is  observed,  subtract  the  augmented  semidiameter.  The  result  is  the 
apparent  altitude  of  the  moon's  centre,  denoted  "  <C's  App.  Alt." 

Apparent  Altitude  of  the  Sun,  Planet,  or  Star.  —  To  the  observed  altitude  apply 
the  index  correction  of  the  sextant,  and  subtract  the  dip,  Table  I. ;  and  if  the  sun  is 
used,  add  its  semidiameter  when  the  lower  limb  is  observed,  or  subtract  it  when  the 
upper  limb  is  observed.  The  result  is  the  apparent  altitude  required,  denoted  by 
"G's  or  #'s  App.  Alt." 

Apparent  Distance. —  1st.  When  the  sun  is  used,  to  the  observed  distance  (cor- 
rected for  index  error  when  necessary)  add  the  moon's  augmented  semidiameter  and 
the  sun's  semidiameter.  2d.  When  a  planet  or  star  is  used,  add  the  moon's  aug- 
mented semidiameter  if  its  nearest  limb  is  observed,  but  subtract  it  if  its  farthest  limb 
is  observed.  The  result  is  "  App.  Dist" 

Moon's  Reduced  Parallax  and  Refraction.  —  Enter  Table  III.  with  the  latitude  of' 
the  place  of  observation  and  the  moon's  horizontal  parallax,  and  take  out  the  correc- 
tion, which  add -to  the  horizontal  parallax.     Call  the  result  the  moon's  reduced  par- 
allax, or  "  €  's  Red.  P." 

Enter  Table  IV.  with  the  moon's  apparent  altitude,  and  take  out  the  mean  reduced 

"  The  sun's  horizontal  parallax  may  be  assumed  as  8".5. 
6 


LUNAR  DISTANCES. 

refraction,  and  apply  to  this  mean  refraction  the  corrections  given  in  Tables  IV.  A. 
and  IV.  B.,  adding  or  subtracting  these  corrections  according  to  the  directions  in  the 
Tables.  The  result  is  the  moon's  reduced  refraction  or  "  d's  Red.  JR." 

Subtract  the  "  C's  Red.  R."  from  the  "  (C's  Red.  P."  and  mark  the  result  as 
""<Ts  Red.  P.  and  JR." 

Reduced  Parallax  and  Refraction  of  Sun,  Planet,  or  Star.*  —  With  the  apparent 
altitude  of  the  sun,  planet,  or  star,  take  from  Table  IV.  the  mean  reduced  refraction, 
which  correct  by  Tables  IV.  A.  and  IV.  B.  If  the  sun  is  observed,  subtract  its  horizontal 
parallax  (which  may  be  always  taken  at  8".5)  from  its  reduced  refraction,  and  mark 
the  result  as  "  Q's  Red.  P.  <$•  £."  If  a  planet  is  observed  subtract  its  horizontal 
parallax,  and  mark  the  result  as  "  jfc's  Red.  P.  fy  R."  If  a  star  is  observed  its 
reduced  refraction  is  at  once  the  required  "  %'s  Red.  P.  fy  B." 

COMPUTATION  OF  THE  TRUE  DISTANCE. 

Take  from  Table  V.  the  four  logarithms  A,  B,  C,  D,t  and  place  these  logs,  each 
at  the  head  of  a  column,  marking  the  columns  A,  B,  C,  and  D,  respectively ;  then 
put  the 

log.  of  <C  's  Red.  P.  <$•  R.          (Table  IX.)          in  Columns  A  and  B 

log.  of  Q's  Red.  P.  fy  R.  "  "         C    "     D 

log.  sine  fs  App.  Alt.      (Bowd.  Table  XXVII.)          "        A    "     D 

log.  sine  Q's  App.  Alt.  "  "         B   "     C 

log.  cotangent  App.  List.  "  "        A    "     C 

log.  cosecant  App.  Dist.  "  "         B    "     D. 

The  sum  of  the  four  logs,  in  Col.  A  is  the  log.  (Table  IX.)  of  the  First  Part  o/<C'f 

Correction,  which  is  to  be  marked  -4-  when  the  App.  Dist.  is  less  than  90°,  but  — 

when  the  App.  Dist.  is  greater  than  90°. 

The  sum  of  the  four  logs,  in  Col.  B  is  the  log.  (Table  IX.)  of  the  Second  Part  of 
(C 's  Correction,  which  is  always  to  be  marked  — . 

The  sum  of  the  four  logs,  in  Col.  C  is  the  log.  (Table  IX.)  of  the  First  Part  of  the 
O's  or  ;fc's  Correction,  which  is  to  be  marked  —  when  the  App.  Dist.  is  less  than 
90°,  but  +  when  the  App.  Dist.  is  greater  than  90°. 

The  sum  of  the  four  logs,  in  Col.  D  is  the  log.  (Table  IX.)  of  the  Second  Part  of 
the  ©'s  or  ^s  Correction,  which  is  always  to  be  marked  -+-. 

Combine  the  first  and  second  parts  of  the  (C 's  correction  according  to  the  signs 
(4-  or  — )  prefixed ;  that  is,  take  their  sum  if  they  have  the  same  sign,  but  their 
difference  if  they  have  different  signs,  and  prefix  the  sign  of  the  greater  to  the 
result,  which  call  "  (C 's  whole  Correction." 

In  the  same  manner  form  the  Q's  or  ?fc's  whole  Correction. 

First  Correction  of  Distance.  —  Combine  the  <£'s  whole  corr.  and  the  0's  or  >Jc'j 

*  The  parallax  of  a  star  being  zero,  its  "  reduced  parallax  and  refraction  "  become,  of  course, 
merely  its  " reduced  refraction";  but  as  no  mistake  can  arise  from  marking  it  as  " ^a  P.  fr  R."  this 
designation  has  been  retained  in  order  to  give  simplicity  and  uniformity  at  once  to  the  rules  and  the 
tables. 

t  No  interpolation  is  required  in  taking  out  these  logarithms. 

7 


CORRECTION   OF 

whole  corr.  according  to  their  signs ;  the  result  is  the  First  Correction  of  Distance, 
which  is  to  be  added  to  or  subtracted  from  the  apparent  distance,  according  as  its 
sign  is  -f-  or  — . 

Second  Correction  of  Distance.  —  Enter  Table  VI.  with  the  Apparent  Distance 
and  the  First  Correction  of  Distance,  and  take  out  the  Second  Correction  of  Distance, 
which  is  to  be  applied  to  the  distance  according  to  the  directions  in  the  side  columns 
of  the  Table. 

Correction  for  the  Elliptical  Figure  of  ike  Moon's  Disc,  or  Contraction  of  ike 
Moon's  Semidiameter  (Table  VII.).  —  Er.'er  Table  VII.  A.  with  the  <C  's  App.  Alt.  and 
<C's  Red.  P.  &  R.,  and  take  out  the  number.  With  this  number  and  the  C's  whole 
correction  enter  Table  VII.  B.  and  take  the  required  contraction,  which  is  to  be  added 
to  the  App.  Dist.  when  the  farthest  limb  is  observed,  but  subtracted  when  the  nearest 
limb  is  observed. 

Correction  for  the  Elliptical  Figure  of  the  Sun's  Disc,  or  Contraction  of  the  Sun's 
Semidiameter  (Table  VIII.).  —  Enter  Table  VIII.  A.  with  the  0's  App.  Alt.  and 
'Q's  Red.  P.  &  R.  and  take  out  the  number.  With  this  number  and  the  0's  whole 
corr.  enter  Table  VIII.  B.  and  take  out  the  required  contraction,  which  is  always  to  be 
subtracted  from  the  distance  (the  nearest  limb  of  the  sun  being  always  observed). 

Correction  for  Compression  or  for  the  Spheroidal  Figure  of  the  Earth.  —  Take 
from  the  Nautical  Almanac  for  the  Greenwich  Date  the  value  of  log.  N,  given  with 
the  lunar  distance  observed.*  To  log.  N  add  the  log.  sine  of  the  latitude  of  the 
place  of  observation  ;  the  sum  is  the  log.  (Table  IX.)  of  the  required  correction  for 
compression.  In  North  latitude  add  this  correction  to  the  distance  if  log.  IV  in  the 
Nautical  Almanac  is  marked  -h,  or  subtract  it  if  log.  N  is  marked  — ;  in  South 
latitude  subtract  the  correction  when  log.  N  is  +,  and  add  it  when  log.  N  is  — . 

All  these  corrections  being  applied  to  the  Apparent  Distance,  the  result  is  the 
True  Distance. 

To  FIND  THE  ERROR  OF  THE  CHRONOMETER. 

Fmd  in  the  Nautical  Almanac  the  two  distances  between  which  the  true  distance 
falls.  Take  out  the  first  of  these  together  with  the  Prop.  Log.  following  it  and  the 
hours  of  Greenwich  time  over  it.  Find  the  difference  between  the  distance  taken 
from  the  Almanac  and  the  true  distance,  and  to  the  log.  of  this  difference  (Table  IX.) 
add  the  Prop.  Log.  from  the  Almanac  ;  the  sum  is  the  log.  (Table  IX.)  of  an  interval 

*  The  values  of  log.  N  are  no  longer  given  in  the  Nautical  Almanac.    For  distances  of 

the  moon  from  the  sun  for  the  year  1855,  they  are  given  in  Table  XI.  of  this  volume.     For 

other  distances,  at  any  time,  log.  N  may  be  found  from  Table  XIL  by  the  following  role  : 

From  the  Nautical  Almanac  take  the  moon's  and  star's  (or  planet's)  declinations  to  the  nearest 

whole  degree.    With  the  moon's  declination  and  apparent  distance  take  from  Table  Xll.  A.  the  fast 

part  of  N,  and  mark  it  with  the  sign  in  the  table  if  the  declination  is  North  ;  but  if  the  declination  is 

South,  change  the  sign  from  -H  to  —  or  from  —  to  -K    With  the  star's  (or  planet's)  declination  and 

apparent  distance,  take  from  Table  XII.  B.  the  second  part  of  N,  and  mark  it  -f-  when  the  declination 

•is  -North,  and  —  when  South.    Take  the  sum,  or  difference,  of  the  two  parts,  according  as  their  signs 

are  the  same  or  different,  and  to  the -resulting  number  prefix  the  sign  of  the  greater.    The  logarithm 

of  this  number  of  seconds,  taken  in  Table  IX.,  with  its  sign  prefixed,  is  the  required  log.  N,  to  be 

used  as  directed  in  the  text. 


LUNAR  DISTANCES. 

of  time  to  be  added  to  the  hours  of  Greenwich  time  taken  from  the  Almanac.  The 
result  is  the  approximate  Greenwich  time. 

To  correct  this  Greenwich  time,  take  the  difference  between  the  two  Prop.  Logs, 
in  the  Almanac  which  stand  against  the  two  distances  between  which  the  true  dis- 
tance falls.  With  this  difference  and  the  interval  of  time  just  found,  enter  Table  X. 
of  this  Appendix,  and  take  out  the  seconds,  which  are  to  be  added  to  the  approx- 
imate Greenwich  time  when  the  Prop.  Logs,  are  decreasing,  but  subtracted  when 
the  Prop.  Logs,  are  increasing.  The  result  is  the  true  Greenwich  time. 

The  difference  between  the  true  Greenwich  time  and  the  time  shown  by  the  chro- 
nometer is  the  error  of  the  chronometer  as  determined  by  the  lunar  observation. 

i 
DEGREE  OF  DEPENDENCE. 

If  the  error  thus  determined  agrees  with  that  deduced  by  means  of  the  rate  ana 
original  error,  the  chronometer  has  run  well,  and  its  rate  is  confirmed ;  if  otherwise, 
more  or  less  doubt  is  thrown  upon  the  chronometer,  according  to  the  degree  of  accu- 
racy of  the  lunar  observation  itself.  An  error  of  10"  in  the  measurement  of  the 
distance  produces  about  208>  error  in  the  Greenwich  time ;  and  since,  even  with  the 
best  observers,  a  single  set  of  distances  is  subject  to  a  possible  error  of  10",  it  may 
be  well  to  consider  the  chronometer  as  still  to  be  trusted  so  long  as  it  does  not  differ 
from  the  lunar  by  more  than  208>.  Since,  however,  so  much  depends  upon  skill  in 
measuring  the  distance,  the  observer  can  only  form  a  correct  judgment  of  the  degree 
of  dependence  to  be  placed  upon  his  own  observations  by  repeated  trials  and  a  care- 
fui  comparison  of  his  several  results. 


CORRECTION   OF 


EXAMPLE  L 

In  Lat.  35°  30'  N.,  Long.  30°  W.,  by  account,  on  September  7th,  1855,  about  6" 
A.  M.,  the  Greenwich  chronometer  showing  8h-  29m-  578>.5  and  supposed  to  be  fast 
21m-  l'-.5  ;  the  observed  distance  of  O's  and  C  's  nearest  limbs  is  43°  52'  10" ;  ob- 
served alt.  _0_  49°  32'  50";  observed  alt.  0  5°  27'  10";  barometer  29.1  inches, 
thermometer  75° ;  height  of  the  eye  above  the  sea  20  feet ;  index  correction  of  the 
sextant  0.  What  is  the  error  of  the  chronometer  on  Greenwich  time  according  to 
these  observations? 

Preparation  of  the  Data* 


Chronometer 
(Fast) 
Greenw.  date  Sept.  6 

h.   m.    B. 
8  29  57.5 
—  21     1.5 

d  's  semid.  N.  A. 
Aug.  Tab.  II. 
d  's  aug.  Semid. 

20    8  56.0 

Obs'd  Alt.  « 
Dip  Tab.  I. 
C  's  aug.  semid. 
C  's  App.  Alt. 

0         1       II 

49  32  50 
—  4  23 
+15     1 
49  43  28 

Obs'd  alt  O 
Dip 
0's  semid 
O'*  App.  Alt. 

d'sRed.R.  Tab.  IV. 
Barom.  Tab.  IV.  A. 
Therm.  Tab.  IV.  B. 
4'sRed.R. 
4'»Red.P. 

i     H 
1  16 
—3 
—4 
1     9 
54  23 

O's  Red.  R.  Tab.  IV. 
Barom.  Tab.  IV.  A. 
Therm.  Tab.  IV.  B. 

©'s  Par. 

4  '«  Red  .  P.  &-  R. 

53  14 

©'s  Red.  P.frR. 

14  50.0 
+11.2 

(t'sPar.N.A. 
Aug.  Tab.  in. 
C's/fed.P. 

54  19.4 
+3.6 

15     1.2 

54  23.0 

O       1        II 

5  27  10 
—  4  23 
-1-15  55 

Obs'd  distance  Ol 
d  's  aug.  semid. 
Q's  semid. 
App.  Dist. 

O        1       I/ 

|d  43  52  10 
+  15    1 
-+-1555 

5  3*  42 

4423    6 

8  57 
—16 
—28 

8  13 

T~5 

Computation  of  the  True  Distance. 


A. 
log.  A.  Tab.  V.                    0.0021 
log.  d  's  Bed.  P.&  R.  TabJX.  3.5043 
log.  sin.  a  's  App.  Alt.         9.8825 
log.  cot.  App.  Dist.               0.0093 

C. 

log.  C.  Tab.  V.                      9.9949 
log.O'sRed.P.&R.  TablX.  2.6857 
log.  sin.  O's  App.  Alt.         8.9929 
log.  cot.  App.  Dist.               0.0093 

App.  Dist 
1st  corr. 
Zdcorr.  Tab.  VI 
Contraction  of  d  '«  ) 
Semid.  Tab.  VII.  5 
Contraction  of  Q's  > 
£ewm/.Tab.VIII.5 
Corr.  for  compression 
True  Distance 

44°  fc  6 
+42  18 
—       16 

0 
—      20 
—        3 

C             log.  Tab.  IX.          3.3982 
\  1st  Part  d  's  corr.        +41  '  42" 

B. 

log.  B.  Tab.  V.                     9.9951 
log.  d  's  Red.  P.  &  R.          3.5043 
log.  sin.  O's  App.  Alt.         8.9929 
log.  cosec.  App.  Dist.           0.1552 

C             log.  Tab.  IX. 
\  1st  Part  O'*  corr. 

D. 

log.  D.  Tab.  V. 
log.  O's  Red.  P  &  R. 
log.  sin.  d  's  App.  Alt 
log.  cosec.  App.  Dist. 
C            log.  Tab.  IX. 
I  2d  Part  Q's  corr. 
0's  whole  corr. 

1.6828 
—0'  48" 

9.9992 
2.6857 
9.8825 
0.1552 

5             log.  Tab.  IX.         2.6475 
>  2d  Part  (I  's  corr.         —  7'  "24" 
O  whole  corr.                +34'  18" 

log.  N.*  Tab.  XL               —0.764 
log.  sin.  Lat  35°  30'  N.      +9.764 
log.  Tab.  EL                     —0.528 

2.7226 
+8'  48" 
+8'    0" 

45    4  45 

*  This  log.  may  also  be  found  from  Table  XII.  by  the  rule  in  the  note  on  p.  8. 


LUNAR  DISTANCES. 


To  find  the  Error  of  the  Chronometer. 


45°  4'  45" 
46     3    17 


P.L.  0.3433    Diff.  P.  logs.  +5 

log.  Tab.  IX.  3.5456 
log.  Tab.  IX.  3.8889 


20 


True  distance 

Distance  N.  A.  at  XVIII> 

Difference 

Approximate  interval 

Add 

Approx.  Gr.  time 

Corr.  Tab.  X. 

True  Gr.  time 

Gr.  time  by  Chronom. 

Chronom.  and  lunar  differ  only 

and  therefore  the  chronometer  may  be  considered  as  going  well. 

This  example,  worked  by  Bowditch's  First  Method,  gives  the  true  distance  45°  & 
44",  differing  from  the  above  59",  in  consequence  of  the  omission  by  Bowditch  of 
the  small  corrections.  This  difference  would  produce  an  error  of  2m*  10s-  in  the 
Greenwich  time,  and  consequently  the  longitude  in  this  case  deduced  by  Bowditch's 
method  would  be  in  error  32'.5 ;  that  is,  more  than  half  a  degree. 


EXAMPLE  II. 

In  Lat.  55°  20'  S.,  Long.  120°  25'  W.,  by  account,  on  August  29th,  1855,  about 
9h-  40ra<  P.  M.,  the  Greenwich  chronometer  showing  5h-  35m-  46'  .2  and  from  previous 
rate  supposed  slow  5m>  12'- ;  the  following  distance  and  altitudes  are  found,  being 
the  mean  of  six  observations  corrected  for  index  errors.  Observed  distance  of  Fo- 
malhaut  and  C  's  farthest  limb  46°  30'  23" ;  observed  alt.  _C_  6°  26'  10" ;  observed 
alt.  Fomalhaut52°  34' 40";  barometer  31  inches;  thermometer  20° ;  height  of  the 
eye  above  the  sea  18  feet.  What  is  the  error  of  the  chronometer  according  to 
these  observations? 


h.   m.    a. 

Chronometer  5  35  46.2 

(Slow)  -H    5  12 

Greenw.  date  Aug.  29d-  17  40  58.2 


Obs'd  Alt.  .1 
Dip  Tab.  I. 
d  's  aug.  semid. 
([^  App.  Alt. 

d's  Red.  R.  Tab.  IV. 
Barom.  Tab.  IV.  A. 
Therm.  Tab.  IV.  B. 
d'sRed.  R. 
CsRed.P. 


O        I       « 

6  26  10 
—  49 

+16  28 


6  38  29 


4-16 
4-32 


8  36 
60  20 


51  44 


Preparation  of  the  Data. 


<Ts  semid.  N.  A. 
Aug.  Tab.  II. 
d  's  aug.  Semid. 

Obs'd  alt.  >fc 

Dip 

#'s  App  Alt. 


#'s  Red.  R.  Tab.  IV. 
Barom.  Tab.  IV.  A. 
Therm.  Tab.  IV.  B. 
#'s  Red.  R. 
#'5  Par. 
#'«  Red.  P.j-R. 
11 


1        H 

16  26.3 
-f-2.0 

(TsPar.N.A. 
Aug.  Tab.  III. 
d'sRed.P. 

66  if* 

-1-8.3 

16  28.3 

60  20.1 

o      /      // 
52  34  40 
—  4     9 
52  30  31 

Obs'd  distance  ;fc 
C  's  aug.  semid. 
App.  Dist. 

<[  1  46  30  23 
—  1628 
46  13  55 

Id 

4-  2 
4-  5 

1  20 
0 

Tao 

CORRECTION   OF  LUNAR  DISTANCES. 


Computation  of  the  True  Distance. 


A. 

log.  A.  Tab.  V.                     0.0274 
log.  «'sRed.P.&R.Tab.IX.3.4919 
log.  sin.  d  's  App.  Alt          9.0632 
log.  cot.  App.  Dist               9.9813 

c. 

log.  C.  Tab.  V.                     9.9999 
logo's  Red.  P.&  R.  TabJX.  1  .9031 
log.  sin.  >f;'s  App.  Alt.         9.8995 
log.  cot.  App.  Dist.               9.9813 
C             log.  Tab.  IX.          1.7838 
i  1st  Part  5|C's  dorr.           —1'  1" 

D. 

log.  D.  Tab.  V.                     0.0267 
log.  #'s  Red.  P.  &  R.          1.9031 
log.  sin.  d  's  App.  Alt          9.0632 
log.  cosec.  App.  Dist.          0.1414 

App.  Dist.                  46  13 
1st  corr.                        —51 
2c?  corr.  Tab.  VI.         — 
Contraction  of  d  's  ) 
Semid.  Tab.  VII.  5   ' 

dorr,  for  romnrfssion     -4- 

55 
32 
22 

17 

14 

C             log.  Tab.  IX. 
<  1st  Part  C  's  corr. 

B. 

log.  B.  Tab.  V. 
log.  d'sKed.P.&R. 
log.  sin.  ^s  App.  Alt. 
log.  cosec.  App.  Dist. 
<             log.  Tab.  IX. 
1  2d  Part  <[  's  corr. 
<l  '«  whole  corr. 

log.  N  • 
log.  sin.  Lat.  55°  S. 

log.  Tab.  IX. 

2.5638 
-J-6'  6" 

0.0001 
3.4919 
9.8995 
0.1414 

3.5329 
—56'  51" 
—50'  15" 

—1.230 
—9.913 

«            log.  Tab.  IX. 
J  2d  Part  >jc's  corr. 
5JC's  whole  corr.  '    ->•  ^ 

1.1344 
+0'  14" 
-M)'  47" 

H-1.143 

True  Distance            45  22 

32 

To  find  the  Error  of  the  Chronometer. 


True  Distance 
Dist.N.A.atXV> 
Difference 
Approx.  interval 
Add 
Approx.  Gr.  time 
Corr.  Tab.  X. 
True  Gr.  time 

45°  22'  32" 
43    51    59 

P.L. 
log.  Tab.  IX. 
log.  Tab.  IX. 

0.2527  Diff.  P.  Logs. 
3.7350 

1    30    33 
2^"  42111'   I** 
15 

3.9877 

17    42      1 

+1 

17    42      2 

Gr.  time  by  chronom.  17    40    58 
Chron.  and  lunar  differ        I      4 

and,  the  distances  having  been  observed  with  care,  the  chronometer  has  probably 
changed  its  rate.  A  second  observation  confirming  this,  we  must,  from  repeated 
lunars,  determine  a  new  rate,  which  may  be  used  until  an  opportunity  occurs  of 
rating  at  a  fixed  place  whose  longitude  is  tolerably  well  known. 

This  example,  worked  by  Bowditch's  Second  Method,  gives  the  true  distance  45° 
2V  31",  which  is  in  error  I7  1",  and  would  produce  in  the  longitude  deduced  from  it 
in  this  case  an  error  of  about  28'. 

*  This  value  of  log.  N.  is  formed  from  Tab.  XII.  by  the  rule  in  the  note  on  page  8  of  this  Appea- 
dix:  tbus 

<C's  dec.  4°  N.,  App.  Dist.  46°  in  Tab.  XII.  A.  gives  1st  Part  of  N  —  1" 
5fc's  dec.  30°  S.,  App.  Dist.  46°  in  Tab.  XII.  B.  gives  2d  Part  ofN— 16 

N-17 
the  log.  of  which  is  in  Tab.  IX.  1.230. 


12 


EXPLANATION    OF  THE  TABLES. 


TABLE  I.  —  Dip  of  the  Sea  Horizon^  computed  by  Delambre's  formula  (Astroi 
mie,  Vol.  III.,  Chap.  XXXVI.),  which,  when  feet  are  substituted  for  metres,  is 

•D  =  58".8 


where  F  =  height  of  the  eye  above  the  level  of  the  sea,  in  feet, 
D  ==  depression  of  the  sea  horizon,  in  seconds. 

TABLE  II.  —  Augmentation  of  the  Moon's  Semidiameter  ^  computed  by  the  formula 
(Francoeur,  Astron.  Pratique,  p.  58) 

x  =  c  s*  sin  h  +  }  c2  s3  sin2  h  4-  £  c2  s3, 

where  h  =  moon's  apparent  altitude, 

s  =  moon's  horizontal  semidiameter, 
x  =  augmentation  of  semidiameter  for  the  altitude  A, 
log  c  =  5.25021. 

TABLE  III.  —  Augmentation  of  the  Moon's  Horizontal  Parallax^  or  correction  to 
reduce  the  moon's  equatorial  horizontal  parallax  to  that  point  of  the  earth's  axis 
which  lies  in  the  vertical  of  the  observer  in  any  given  latitude,  computed  by  the 
formulas 

A,r=,r(J-l),     i=y  (1_J,  s.n2(/)), 

where  w  =-  equatorial  horizontal  parallax, 
<f>  =  latitude, 

e  =  eccentricity  of  the  meridian,  log  e3  =  7.81602, 
A  TT  =  augmentation  of  the  horizontal  parallax  for  the  latitude  cj>. 

TABLE  IV.  —  Mean  Reduced  Refraction  for  Lunars^  computed  by  the  formula 


- 

cos  h       sin  h  ' 

*  Tables  I.,  II.,  and  IX.,  which  are  not  peculiar  to  my  method,  are  inserted  to  save  a  reference  to 
other  works  for  them  ;  so  that  the  computer  requires,  in  connection  with  this  volume,  only  a  table  of 
logarithmic  sines  and  tangents.  These  three  tables  have,  however,  been  recomputed,  as  stated  in  the 
u  Explanation." 

13 


EXPLANATION 

where  h  =  the  apparent  altitude, 

r  =  mean  refraction,  barometer  30  inches,  Fahrenheit's  thermometer  50°, 
r'  =  mean  reduced  refraction  for  lunars. 

The  refractions  employed  are  BESSEL'S,  and  were  taken  from  his  table  (Astro* 
nomische  Untersuchungen,  Vol.  I.  p.  200),  which  gives  directly  K  =z  r  tan  h. 

TABLES  IV.  A.  AND  IV.  B.  —  Corrections  of  the  Mean  Refraction  for  the  Height 
of  the  Barometer  and  Thermometer,  deduced  also  from  BessePs  table  above  cited. 
These  tables  serve  for  correcting  either  r  or  r',  according  as  the  one  or  the  other  is 
taken  as  the  argument. 

TABLE  V.  —  Logs,  of  A,  B,  C,  and  D,  for  computing  the  First  Correction  of  the 
Lunar  Distance,  computed  by  the  formulas 


sin  A          • 

p_  ^sin^g-Aff)          n__ 
' 


sin  2  h        ' 

where  h  =  moon's  apparent  altitude, 

H  =  sun's,  planet's,  or  star's  apparent  altitude  (denoted  in  the  tables  by  O  '• 

or  ;fc's  App.  Alt.), 

A  h  =  difference  of  <C  's  apparent  and  true  altitudes, 
A  H  =  difference  of  Q's  or  jfc's  apparent  and  true  altitudes, 
log  K'  =  .000126, 

and  A  A,  A  H  were  computed  from  the  arguments  "  apparent  altitude  "  and  "  re- 
duced parallax  and  refraction  "  by  the  formulas 

A  h  =  (p  —  r1)  K'  cos  h,     &H=(R'  —  P)  cos  JJ, 

where  p  —  r'  =  <C  's  reduced  parallax  and  refraction, 

R'  —  P==  O's  °r  sfc's  reduced  parallax  and  refraction, 
p  =  <£  's  horizontal  parallax  -f-  A  tr  (Table  III.), 
P  =  O's  or  j|c's  horizontal  parallax  (for  a  star  P  =  0), 
r'  =  C's  reduced  refraction  (Table  IV.), 
R'  =  O's  or  #'s  reduced  refraction  (Table  IV.). 

When  h  and  H  become  90°,  the  values  of  B  and  D  assume  the  indeterminate 
forms 

R      °          n      ° 

^  =  6'     J)=6'     . 

and  therefore,  for  computing  their  logarithms  near  the  end  of  the  table,  the  formulas 
were  transformed  as  follows  : 

B  =  C  K1  [1-*-  £  (R1  —  P)  sin  1"  sin  H]  , 

D  =  ^  [1  —  £  (p  —  r')  K1  sin  1"  sin  h]  . 

By  means  of  Logs.  A  ,  J5,  C,  and  D,  the  first  correction  of  the  distance  is  found  by 
the  following  formulas  :  — 

14 


OF  THE  TABLES. 

if 

d  =  apparent  distance  of  C  from  0  or  #., 
A!  d  =  first  correction, 
A'  =       A  (p  —  r')  sm  h  cot  d, 
B'  =  —  B  (p  —  r')  sin  Jfcosec  d, 
C'  =  —  C  ( R'  —  P)  sin  H  cot  d, 
D'  =       U  (£'  —  P)  sin  A  cosec  d, 
then 

A1d  =  (4'-h.B')-4-(C'  +  j)/), 

and  in  the  directions  for  using  the  tables,  A1  -+-  B1  is  called  the  "  whole  correction 
for  <£,"  and  C'  4-  D1  the  "  whole  correction  for  0  or  #." 

TABLE  VI.  —  Second  Correction  of  the  Lunar  Distance^  computed  by  the  formula 

&z  d  =  —  J  A!  <F  sin  I"  cot  d. 
Strictly,  this  formula  should  be 

AS  d  =  —  £  A  &  sin  1"  cot  d, 

where  A  <Z  =  whole  correction  of  distance  =  A!  e£  -H  A2  d ,  so  that  by  entering  the 
table  first  with  At  rf  we  find  only  an  approximate  value  of  A2  d.  But  if  with  this  ap- 
proximate value  we  form  the  approximate  whole  correction  A!  d  -f-  A3  d,  and  enter 
with  this  as  the  argument  in  the  place  of  Aj  d,  we  have  the  true  value  of  A2  d.  It  is 
evident,  however,  from  the  table  itself,  that,  in  most  practical  cases,  this  degree  of 
precision  is  unnecessary. 

TABLE  VII.  —  For  finding  the  Correction  of  the  Lunar  Distance  for  the  Contrac* 
tion  of  the  Moon's  Semidiameter  (by  refraction),  computed  by  the  formula. 

A  s  =  A  s0  ^_r,)2cos2£ 
where  the  notation  already  employed  is  preserved,  and 

A  $0  =  contraction  of  the  moon's  vertical  semidiameter  at  the  altitude  h, 
A  *  =  contraction  of  the  inclined  semidiameter,  or  of  that  which  lies  in  the  dL 
rection  of  the  lunar  distance. 

This  table  is  subdivided  into  Tables  VII.  A.  and  VII.  B.     If  we  put 

*  =  (p  — 
then 

A*  =  (4'-i-#)8xC 

where  /  is  an  arbitrary  factor  employed  to  give  g  convenient  integral  valuesr  and  m 
these  tables  we  have  taken  /  =  18000000 ;  A  SQ  ,  A  s ,  p  —  r',  and  A1  -f-  B1  being 
all  expressed  in  seconds.  Table  VII.  A.  gives  the  value  of  g  with  the  arguments 
p  —  r1  and  h ;  and  Table  VII.  B.  gives  A  s  with  the  arguments  A!  -H  B'  and  g. 

To  find  A  s0  from  the  arguments  p  —  r'  and  7t,  the  mean  value  of  r'  for  the  altitude 
A  was  added  to  p  —  r',  so  that  p  became  known ;  whence  also  the  semidiameter,  — 

15 


EXPLANATION 

and  consequently,  by  .neans  of  the  refraction  tables,  the  contraction  A  SQ.  The  value 
of  A  s  given  by  Table  VII.  B.  is  therefore  that  which  corresponds  to  a  mean  state  of 
the  air,  and  in  extreme  cases  may  be  in  error  4",  but  in  no  probattle  case  will  it  be  in 
error  more  than  1".  The  true  value  of  A  s  might,  however,  always  be  found  by  cor- 
recting it,  as  refraction,  by  Tables  IV.  A.  and  IV.  B. 

TABLE  VIII.  —  For  finding  the  Correction  of  the  Lunar  Distance  for  the  Contra^" 
lion  of  the  Sun's  Semidiameter  (by  refraction),  computed  by  the  formula 


where,  in  addition  to  the  notation  already  employed,  A  SQ  =  contraction  of  the  sun's 
vertical  semidiameter  at  the  altitude  H;  A  ,  S  =  contraction  of  the  inclined  semi- 
diameter.  Table  VIII.  A.,  with  the  arguments  R'  —  P  and  H,  gives  a  number 


in  which  A  S0  is  taken  for  a  mean  value  of  the  sun's  semidiameter,  P  is  assumed  al 
8".5,  and  F  is  an  arbitrary  factor  =  yfo  ;  jR'  —  P,  C'  -4-  JDf,  A  S0,  and  A  S  being  all 
expressed  in  seconds.  Table  VIII.  B.,  with  the  arguments  O  -+•  D1  and  G,  gives 


This  value  of  A  S  is  that  which  belongs  to  the  actual  state  of  the  air,  but  is  fjbr  a 
mean  value  of  the  sun's  semidiameter,  the  variations  of  which  would  not  chang§  the 
tabular  value  by  more  than  0".5  in  any  case.  \ 

*\   »<•  ;  '\  *-**•   '  '•  ' 

TABLE  IX.  —  Logarithms  of  Seconds.      This  table  contains  the*  common  loga-    , 
rithm  answering  to  the  arc  (either  space  or  time)  in  the  argument.     For  the  con- 
venience of  the  navigator,  this  table  is  given  with  the  argument,  degrees,  minutes* 
and  seconds,  or  hours,  minutes,  and  seconds,  for  every  ten  seconds,  at  the  side,  the^ 
unit  figure  of  the  seconds  being  found  at  tba  top. 

The  logarithm  is  given  with  the  proper  ?  caracteristic  prefixed.  k£ 

TABLE  X.  —  This  table  contains  the  correction  for  second  differences  of  the  moon's 
motion. 

TABLE  XI.  contains  the  values  of  log.  N  for  the  distances  from  the  sun,  com- 
puted by  the  following  formula  :  — 

-^\; 
sin  d        tan  d  / 

where  «•  =  <C  's  equatorial  horizontal  parallax, 
A  =  sun's  declination, 
d  =  <L  's  declination, 
d  =  angular  distance  of  the  moon  and  sun  referred  to  the  centre  of  the  earth, 

Q  .  ^ 

JV'  =  —r,  -  o  •  o  .v  i  f°r  which  the  value  of  N1  is  taken  corresponding  to  - 
«/(l  —  e^  sin-*  <p) 

<£  =  45°,  or  log.  N'  =  7.8170. 
16 


OF  THE  TABLES. 

TABLE  XII.  —  For  finding  the  Value  of  N.  By  this  table  an  approximate  value 
of  IV  may  be  readily  found,  with  sufficient  accuracy  for  ordinary  purposes.  When 
very  great  accuracy  is  desired,  the  formula  given  in  the  explanation  of  Table  XI. 
should  be  used.  In  Table  XII.  the  moon's  horizontal  parallax  is  assumed  at  its 
mean  value  ==  57'  30",  and  the  two  parts  of  IV  are  separately  tabulated  by  the 
formulas 

a  (=  first  part  of  IV)    =  —  IV'  TT  sin  S  cotan  d, 

I  (=  second  part  of  IV)  =       N1  it  sin  A  cosec  d  ; 
from  which  IV  is  found  by  the  formula 


NOTE.  —  The  demonstration  of  most  of  the  formulas  cited  in  the  preceding  explanation  is  girea  IB 
the  Astronomical  Journal  (Cambridge,  Mass.)}  Vol.  II.,  $Tos.  3  and  4. 


17 


TABLES. 


TABLE  I. 

TABLE  II. 

TABLE  III. 

Dip  of  the  Sea 
Horizon. 

Augmentation  of  the  Moon's  Semidiaraeter. 

Augmentation  of  the 
Moon's  Hor.  Parallax. 

Annar- 

}>'s  Semidiameter. 

Lati- 

}>'s Horizontal 

Height 
of  the 

Dip  of  the 
Horizon. 

Af^JBI 

ent 
Altitude 

14' 

15' 

16' 

17' 

tude 
ofOb- 
serva- 

Parallax. 

Eye. 

of  J). 

30" 

0" 

30" 

0" 

30" 

0" 

tion. 

53' 

57' 

61' 

Feet. 

If 

n 

n 

/< 

n 

n 

o 

•  n 

I 

g 

0 

0  00 

0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0 

0.0 

0.0 

0.0 

1 

0  59 

2 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

2 

0.0 

0.0 

0.0 

2 

1  23 

4 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

4 

0.1 

0.1 

0.1 

3 

1  42 

6 

1.5 

1.6 

1.7 

1.9 

2.0 

2.1 

6 

0.1 

0.1 

0.1 

4 

1  58 

8 

2.0 

2.1 

2.3 

2.4 

2.6 

2.7 

8 

0.2 

0.2 

0.2 

5 

2  11 

10 

2.4 

2.6 

2.8 

3.0 

3.2 

3.4 

10 

0.3 

0.3 

0.4 

6 

2  24 

12 

2.9 

3.1 

3.3 

3.6 

3.8 

4.0 

12 

0.5 

0.5 

0.5 

7 

2  36 

14 

3.4 

3.6 

3.9 

4.1 

4.4 

4.7 

14 

0.6 

0.7 

0.7 

8 

246 

16 

3.8 

4.1 

4.4 

4.7 

5.0 

5.3 

16 

0.8 

0.9 

0.9 

9 

2  56 

18 

4.3 

4.6 

4.9 

5.2 

5.6 

5.9 

18 

1.0 

1.1 

1.1 

10 

306 

20 

4.7 

5.1 

5.4 

5.8 

6.1 

6.5 

20 

1.2 

1.3 

1.4 

11 

3  15 

22 

5.2 

5.5 

5.9 

6.3 

6.7 

7.1 

22 

1.5 

1.6 

1.7 

12 

3  24 

24 

5.6 

6.0 

6.4 

6.8 

7.3 

7.7 

24 

1.7 

1.9 

2.0 

13 

3  32 

26 

6.0 

6.5 

6.9 

7.4 

7.8 

8.3 

26 

2.0 

2.2 

2.3 

14 

3  40 

28 

6.5 

6.9 

7.4 

7.9 

8.4 

8.9 

28 

2.3 

2.5 

2.6 

15 

3  48 

30 

6.9 

7.3 

7.9 

8.4 

8.9 

9.5 

30 

2.6 

2.8 

3.0 

16 

3  55 

32 

7.3 

7.8 

8.3 

8.9 

9.4 

10.0 

32 

2.9 

3.1 

3.4 

17 

4  02 

34 

7.7 

8.2 

8.8 

9.4 

10.0 

10.6 

34 

3.3 

3.5 

3.8 

18 

09 

36 

8.1 

8.6 

92 

9.8 

10.5 

11.1 

36 

3.6 

3.9 

4.1 

19 

16 

38 

8.4 

9.0 

97 

10.3 

10.9 

11.6 

38 

4.0 

4.3 

4.6 

20 

23 

40 

8.8 

9.4 

10.1 

10.7 

11.4 

12.1 

40 

4.3 

4.6 

5.0 

21 

29 

42 

9.2 

9.8 

10.5 

11.2 

11.9 

12.6 

42 

4.7 

5.0 

5.4 

22 

36 

44 

9.5 

10.2 

10.9 

11.6 

12.3 

13.1 

44 

5.0 

5.4 

5.8 

23 

42 

46 

9.8 

10.5 

11.3 

12.0 

12.8 

13.6 

46 

5.4 

5.8 

6.2 

24 

48 

48 

10.2 

10.9 

11.6 

12.4 

13.2 

14.0 

48 

5.8 

6.2 

6.6 

25 

4  54 

50 

10.5 

11.2 

12.0 

12.8 

13.6 

14.4 

Ufr 

6.1 

6.6 

7.1 

26 

5  00 

52 

10.8 

11.5 

12.3 

13.1 

14.0 

14.9 

52 

6.5 

7.0 

7.5 

27 

5  06 

54 

11.1 

11.8 

12.7 

13.5 

14.4 

15.3 

54 

6.8 

7.4 

7.9 

28 

5  11 

56 

11.3 

12.1 

13.0 

13.8 

14.7 

15.6 

56 

7.2 

7.7 

8.3 

29 

5  17 

58 

11.6 

12.4 

13.3 

141 

15.1 

16.0 

58 

7.5 

8.1 

8.6 

30 

5  22 

60 

11.8 

12.7 

i3.5 

14.4 

15.4 

16.3 

60 

7.8 

8.4 

9.0 

35 

5  48 

62 

12.1 

12.9 

13.8 

14.7 

15.7 

16.6 

62 

8.1 

8.8 

.9.4     ; 

40 

6  12 

64 

12.3 

13.2 

14.1 

15.0 

16.0 

16.9 

64 

8.4 

9.1 

9.7 

45 

6  34 

66 

12.5 

13.4 

14.3 

15.2 

16.2 

17.2 

66 

8.7 

9.4 

10.0 

50 

6  56 

68 

12.7 

13.6 

14.5 

15.5 

16.5 

17.5 

68 

9.0 

9.7 

10.3 

55 

7  16 

70 

12.9 

13.8 

14.7 

15.7 

16.7 

17.7 

70 

9.2 

9.9 

10.6 

60 

7  35 

72 

13.0 

13.9 

14.9 

15.9 

16.9 

17.9 

72 

9.5 

10.2 

10.9 

65 

7  54 

74 

13.1 

14.1 

15.0 

16.0 

17.1 

18.1 

74 

9.7 

10.4 

11.1 

70 

8  12 

76 

13.3 

14.2 

15.2 

16.2 

17.2 

18.3 

76 

9.8 

10.6 

11.3 

75 

8  29 

78 

13.4 

14.3 

15.3 

16.3 

17.4 

18.4 

78 

10.0 

10.8 

11.5 

80 

846 

80 

13.5 

14.4 

15.4 

16.4 

17.5 

18.6 

80 

10.1 

10.9 

11.7 

85 

9  02 

82 

13.5 

14.5 

15.5 

16.5 

17.6 

18.7 

82 

10.3 

11.0 

11.8 

90 

9  18 

84 

136 

14.6 

15.6 

16.6 

17.6 

18.7 

84 

10.3 

11.1 

11.9 

95 

9  33 

86 

13.6 

14.6 

15.6 

16.6 

17.7 

18.8 

86 

10.4 

11.2 

12.0 

100 

9  48 

88 

13.7 

14.6 

15.6 

16.7 

17.7 

18.8 

88 

10.4 

11.2 

12.0 

90 

13.7 

14.6 

15.6 

16.7 

17.7 

18.8 

90 

10.5 

11.3 

12.0 

19 


TABLE    IV. 


Mean  Reduced  Refraction  for  Lunars.     Barometer  30  inches.    Fahrenheit's  Thermometer  fiCR 


Apparent 
Altitude. 

Reduced 
Refraction. 

Biff, 
to    '. 

Apparent 
Altitude. 

Reduced 
Refraction. 

Apparent 
Altitude. 

Reduced 
Refraction. 

Apparent 
Altitude. 

Reduced 
Refraction. 

O          1 

5       0 

i        H 
9     54.2 

(/ 
.6 

O          1 

10       0 

i        n 
5     24.1 

o       / 
15       0 

i        // 
3     41.7 

0 

27       0 

2       7.S 

5 

9     46.3 

.5 

5 

5     21.6 

10 

3     39.4 

27     30 

2       5.7 

10 

9     38.6 

.5 

10 

5     19.2 

20 

3     37.1 

28       0 

2       3.7 

15 

9     31.0 

.5 

15 

5     16.8 

30 

3     34.9 

28     30 

2       1.7 

20 

9     23.7 

.4 

20 

5     14.4 

40 

3     32.7 

29       0 

59.8 

25 

9     16.5 

1.4 

25 

5     12.1 

50 

3     30.6 

29     30 

58.0 

5     30 

9       9.5 

1.4 

10    30 

5       9.8 

16       0 

3     28.5 

30       0 

56.2 

35 

9       2.7 

1.3 

35 

5       7.5 

10 

3     26.5 

30     30 

54.5 

40 

8     56.0 

1.3 

40 

5       5.3 

20 

3     24.5 

31       0 

52.8 

45 

8     49.5 

1.3 

45 

5       3.1 

30 

3     22.6 

31     30 

51.2 

50 

8    43.1 

1.2 

50 

5       0.9 

40 

3     20.7 

32       0 

1     49.7 

55 

8     36.9 

1.2 

55 

4     58.8 

50 

3     18.8 

32     30 

1     48.2 

6       0 

8    309 

1.2 

11       0 

4    56.7 

17       0 

3     16.9 

33       0 

1     46.7 

5 

8     24.9 

1.2 

5 

4     54.6 

10 

3     15.1 

33     30 

1     45.3 

10 

8     19.1 

1.1 

10 

4     52.5 

20 

3     13.4 

34       0 

1     44.0 

15 

8     13.4 

1.1 

15 

4     50.5 

30 

3     11.6 

34     30 

1     42.7 

20 

8      7.8 

1.1 

20 

4     48.5 

40 

3       9.9 

35       0 

1     41.4 

25 

8      2.4 

1.1 

25 

4     46.6 

50 

3       8.2 

35     30 

1     40.2 

6     30 

7    57.0 

1.0 

11     30 

4    44.6 

18       0 

3       6.6 

36       0 

1     39.0 

35 

7     51.8 

1.0 

35 

4     42.7 

10 

3       5.0 

36     30 

1     37.8 

40 

7    46.7 

1.0 

40 

4     40.8 

20 

3       3.4 

37       0 

36.7 

45 

7    41.7 

1.0 

45 

4     38.9 

30 

3       1.8 

37     30 

35.6 

50 

7    36.7 

1.0 

50 

4     37.1 

40 

3       0.3 

38       0 

34.5 

55 

7    31.9 

0.9 

55 

4    35.3 

5C 

2     58.8 

38     30 

33.5 

7      0 

7    27.2 

0.9 

12       0 

4     33.5 

19       0 

2     57.3 

39       0 

32.5 

5 

7     22.6 

0.9 

5 

4     31.7 

10 

2     55.9 

39     30 

31.5 

10 

7     18.0 

0.9 

10 

4     30.0 

20 

2     54.4 

40       0 

30.6 

15 

7     13.6 

0.9 

15 

4     28.3 

30 

2     53.0 

40     30 

29.6 

20 

7      9.2 

0.9 

20 

4     26.6 

40 

2     51.6 

41       0 

28.7 

25 

7      4.9 

0.8 

25 

4     24.9 

50 

2     50.3 

41     30 

27.8 

7    30 

7      0.8 

0.8 

12    30 

4     23.2 

20       0 

2     49.0 

42       0 

27.0 

35 

6     56.6 

0.8 

35 

•  4     21.6 

10 

2     47.6 

42     30 

26.2 

40 

6     52.6 

0.8 

40 

4     20.0 

20 

2     46.4 

43       0 

25.4 

45 

6    48.6 

0.8 

45 

4     18.4 

30 

2     45.1 

43     30 

24.6 

50 

6    44.8 

0.8 

50 

4     16.8 

40 

2     43.8 

44       0 

23.8 

55 

6    40.9 

0.7 

55 

4     15.2 

50 

2     42.6 

44     30 

23.1 

8       0 

6    37.2 

0.7 

13       0 

4     13.7 

21       0 

2     41.4 

45       0 

22.4 

5 

6     33.5 

0.7 

5 

4     12.2 

10 

2     40.2 

46       0 

21.0 

10 

6     29.9 

0.7 

10 

4     10.7 

20 

2     39.0 

47       0 

19.6 

15 

6     26.3 

0.7 

15 

4       9.2 

30 

2     37.9 

48       0 

18.4 

20 

6     22.8 

0.7 

20 

4      7.7 

40 

2     36.7 

49       0 

17.2 

25 

6     19.4 

0.7 

25 

4       6.3 

50 

2     35.6 

50       0 

16.0 

8     30 

6     16.0 

0.7 

13     30 

4       4.8 

22       0 

2     34.5 

51       0 

15.0 

*  35 

6     12.7 

0.6 

35 

4       3.4 

10 

2     33.4 

52       0 

13.9 

40 

6       9.5 

0.6 

40 

4       2.0 

20 

2     32.4 

53       0 

13.0 

45 

6       6.3 

0.6 

45 

4      0.6 

30 

2     31.3 

54       0 

12.0 

50 

6       3.1 

0.6 

50 

3     59.3 

40 

2     30.3 

55       0 

11.1 

55 

6       0.0 

0.6 

55 

3     57.9 

50 

2     29.2 

56       0 

10.3 

9      0 

5    57.0 

0.6 

14      0 

3     56.6 

23       0 

2     28.2 

57       0 

9.5 

5 

5     54.0 

0.6 

5 

3     55.3 

20 

2     26.3 

58       0 

8.7 

10 

5     51.1 

0.6 

10 

3     54.0 

40 

2     24.4 

59       0 

8.0 

15 

5     48.2 

0.6 

15 

3     52.7 

24      0 

2     22.5 

60       0 

7.3 

20 

5     45.3 

0.6 

20 

3     51.4 

20 

2     20.7 

62       0 

6.0 

25 

5     42.5 

0.5 

25 

3     50.1 

40 

2     18.9 

64       0 

4.9 

9    30 

5     39.8 

0.5 

14    30 

3     48.9 

25       0 

2     17.2 

66       0 

3.8 

35 

5     37.0 

0.5 

35 

3     47.6 

20 

2     15.5 

68       0 

2.9 

40 

5     34.4 

0.5 

40 

3     46.4 

40 

2     13.9 

70       0 

2.0 

45 

5     31.7 

0.5 

45 

3     45.2 

26       0 

2     12.3 

73       0 

1.0 

50 

5     29.2 

0.5 

50 

3     44.0 

20 

2     10.8 

76       0 

0.1 

55 

5     26.6 

0.5 

55 

3     42.8 

40 

2       9.3 

80       0 

0     59.2 

10      0 

5     24.1 

15      0 

3     41.7 

27       0 

2      7.8 

90      0 

0     58.3 

20 


TABLE    IV.    A. 

Correction  of  the  Mean  Refraction  for  the  Height  of  the  Barometer. 


Barometer. 

MEAN     REFRACTION. 

Barometer. 

0' 

1' 

2' 

3' 

4' 

5' 

6' 

V 

8' 

9' 

1O 

Subtract. 

0 

3('i 

0 

34 

0 

30 

6 

3tt 

o 

3fi 

i 

30 

6' 

30 

0    30' 

,!, 

34 

0 

30^ 

t 

Add. 

27.50 

48 

a 
51 

0 

2 

5 

7 

10 

12 

15 

17 

20 

23 

25 

28 

30 

33 

35 

38 

40 

43 

45 

27.55 

0 

2 

9 

7 

10 

12 

15 

17 

20 

22 

25 

27 

80 

32 

35  37 

40 

42 

45 

47 

50 

27.60 

0 

2 

5 

7 

10 

12 

14 

17 

19 

22 

24 

27 

29 

31 

34  36 

39 

41 

44 

46 

49 

27.65 

0 

2 

5 

7 

9 

12 

14 

It) 

19 

21 

24 

26 

28  31 

33  '36 

38 

40 

43 

45 

48 

27.70 

0 

2 

5 

7 

9 

11 

14 

16 

18 

21 

23 

25 

28 

30 

32 

35 

.37 

39 

42 

44 

47 

27.75 

0 

2 

4 

7 

9 

11 

13 

16 

18 

20 

2.3 

25 

27 

29 

32 

34 

36 

39 

41 

43 

46 

27.80 

0 

2 

4 

7 

9 

11 

13 

15 

18 

20 

22 

24 

27 

29 

31  33 

35 

,38 

40 

42 

45 

27.85 

0 

2 

4 

6 

9 

11 

13 

15 

17 

19 

22 

24 

26 

28 

30,32 

35 

.37 

39 

41 

44 

27.90 

0 

2 

4 

6 

8 

10 

13 

15 

17 

19 

21 

23 

25 

27 

3032 

34 

36 

38 

40 

43 

27.95 

0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

21 

23 

25 

27 

29 

31 

33 

35 

-37 

39 

42 

28.00 

0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

,38 

41 

28.05 

0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

25 

27 

29 

31 

33 

35 

37 

39 

28.10 

0 

2 

4 

6 

8 

9 

11 

13 

15 

17 

19 

21 

23 

25 

27 

29 

31 

33 

34 

36 

38 

28.15 

Q 

2 

4 

6 

7 

9 

11 

13 

15 

17 

19 

20 

22 

24 

26 

28 

30 

32 

34 

36 

37 

28.20 

0 

2 

4 

5 

7 

9 

11 

13 

14 

16 

18 

20 

22 

24 

25 

27 

29 

31 

33 

35 

36 

28.25 

0 

2 

3 

5 

7 

9 

10 

12 

14 

16 

18 

19 

21 

23 

25 

26 

28 

30 

32 

34. 

35 

28.30 

0 

2 

3 

5 

7 

8 

10 

12 

14 

15 

17 

19 

21 

22 

24 

26 

27 

29 

31 

33 

34 

28.35 

0 

2 

3 

5 

7 

8 

10 

12 

13 

15 

17 

18 

20 

22 

23 

25 

27 

28 

30 

32 

33 

28.40 

0 

2 

3 

5 

6 

8 

10 

11 

13 

14 

16 

18 

19 

21 

23 

24 

26 

27 

21J 

31 

32 

28.45 

0 

2 

3 

5 

G 

8 

9 

11 

12 

14 

16 

17 

19 

20 

22 

23 

25 

27 

28 

30 

31 

28.50 

0 

1 

3 

4 

6 

7 

9 

10 

12 

14 

15 

17 

18 

20 

21 

23 

24 

26 

27 

29 

30 

31.50 

28.55 

0 

1 

3 

4 

6 

7 

9 

10 

12 

13 

15 

16 

17 

19 

20 

22 

23 

25 

26 

28 

29 

31.45 

28.60 

0 

1 

3 

4 

6 

7 

8 

10 

11 

13 

14 

15 

17 

18 

20 

21 

23 

24 

25 

27 

28 

31.40 

28.65 

0 

1 

3 

4 

5 

7 

8 

9 

11 

12 

14 

15 

16 

18 

19 

20 

22 

23 

25 

26 

27 

31.35 

28.70 

0 

1 

3 

4 

5 

6 

8 

9 

10 

12 

13 

14 

16 

17 

18 

20 

21 

22 

24 

25 

26 

31.30 

28.75 

0 

1 

2 

4 

5 

6 

7 

9 

10 

11 

13 

14 

15 

16 

18 

19 

20 

21 

23 

24 

25 

31.25 

28.80 

0 

1 

2 

4 

5 

6 

7 

8 

10 

11 

12 

13 

14 

16 

17 

18 

19 

21 

22 

23 

24 

31.20 

2885 

0 

1 

2 

3 

5 

6 

7 

8 

9 

10 

12 

13 

14 

15 

16 

17 

19 

20 

21 

22 

23 

31.15 

28.90 

0 

1 

2 

3 

4 

5 

7 

8 

9 

10 

11 

12 

13 

14 

16 

17 

18 

19 

20 

21 

22 

31.10 

28.95 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

31.05 

29.00 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

31.00 

29.05 

0 

1 

2 

3 

4 

5 

8 

7 

8 

9 

10 

11 

11 

12 

13 

14 

15 

16 

17 

18 

19 

30.95 

29.10 

0 

1 

2 

3 

4 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

15 

16 

17 

18 

30.90 

29.15 

0 

1 

2 

3 

3 

4 

5 

6 

7 

8 

9 

9 

10 

11 

12 

13 

14 

15 

15 

16 

17 

30.85 

29.20 

0 

1 

2 

2 

3 

4 

5 

6 

6 

7 

8 

9 

10 

10 

11 

12 

13 

14 

15 

15 

16 

30.80 

29.25 

0 

2 

3 

4 

4 

5 

6 

7 

8 

8 

9 

10 

11 

11 

12 

13 

14 

14 

15 

30.75 

29.30 

0 

2 

3 

3 

4 

5 

6 

6 

7 

8 

8 

9 

10 

11 

11 

12 

13 

13 

14 

30.70 

29.35 

0 

2 

3 

3 

4 

5 

5 

6 

7 

7 

8 

9 

9 

10 

10 

11 

12 

13 

13 

30.65 

29.40 

0 

2 

2 

3 

4 

4 

5 

5 

6 

7 

7 

8 

8 

9 

10 

10 

11 

12 

12 

30.60 

29.45 

0 

2 

2 

3 

3 

4 

4 

5 

6 

6 

7 

7 

8 

8 

9 

9 

10 

11 

11 

30.55 

29.50 

0 

0 

2 

2 

3 

3 

4 

5 

5 

6 

6 

7 

7 

8 

8 

9 

9 

10 

10 

30.50 

29.55 

0 

0 

2 

2 

3 

3 

4 

4 

5 

5 

5 

6 

6 

7 

7 

8 

8 

9 

9 

30.45 

29.60 

0 

0 

2 

2 

2 

3 

3 

4 

4 

4 

5 

5 

6 

6 

6 

7 

7 

8 

8 

30.40 

29.65 

0 

0 

1 

2 

2 

2 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

7 

7 

30.35 

29.70 

0 

0 

1 

1 

2 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

5 

5 

6 

6 

30.30 

29.75 

0 

0 

0 

1 

1 

1 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

4 

5 

5 

5 

30.25 

29.80 

0 

0 

0 

1 

1 

1 

1 

2 

2 

2 

2 

2 

3 

3 

3 

3 

3 

4 

4 

4 

30.20 

29.85 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

30.15 

29.90 

0 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

2 

30.10 

29.95 

0 

0 

0 

0 

0 

0 

.  0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

30.05 

30.00 

0 

0 

0 

0 

0 

0 

0 

0 

00 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

30.00 

Subtract. 

0 

30' 

0 

30 

& 

30 

0 

30' 

0 

30 

0 

30 

6' 

3fl 

1 

30' 

6' 

30' 

i 

36 

o' 

Add. 

0' 

1' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

10' 

MEAN     REFRACTION. 

21 


TABLE    IV.    B. 

Correction  of  the  Mean  Refraction  for  the  Height  of  the  Thermometer. 


Thermora. 

MEAN  REFRACTION. 

Thermom. 

O' 

1' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

10 

Add. 

6 

36 

0 

30' 

I 

36 

6' 

30' 

t 

3ft 

6 

30 

6' 

3# 

6' 

3# 

6' 

sfi 

&  3ft 

6' 

Add. 

Q 

—10 

0 

4 

8 

12 

16 

20 

24 

28 

33 

37 

41 

46 

50 

55 

60 

65 

70 

75 

80  85 

90 

—10 

—  8 

0 

4 

8 

12 

15 

19 

23 

27 

31 

36 

40 

44 

48 

53 

58  62 

67 

72 

77:82 

87 

—  8 

—  6 

0 

4 

7 

11 

15 

19 

22 

26 

30 

34 

38 

42 

47 

51 

55  60 

64 

69 

74J79 

84 

—  6 

i  t   A 

0 

4 

7 

11 

14 

18 

22 

25 

29 

33 

37 

41 

45 

49 

53 

57 

62 

66 

71 

76 

80 

—  4 

—  2 

0 

3 

T 

10 

14 

17 

21 

24 

28 

31 

35 

39 

43 

47 

51 

55 

59 

64 

68 

72 

77 

—  2 

0 

0 

3 

7 

10 

13 

16 

20 

23 

27 

30 

34 

37 

41 

45 

49 

53 

57 

61 

65 

69 

74 

0 

2 

0 

a 

6 

9 

12 

16 

19 

22 

25 

29 

32 

36 

39 

43 

47 

50 

54 

58 

62  66 

70 

2 

4 

0 

3 

6 

9 

12 

15 

18 

21 

24 

28 

31 

34 

37 

41 

44 

48 

52 

55 

59 

63 

67 

4 

6 

0 

3 

6 

8 

11 

14 

17 

20 

23 

26 

29 

32 

36 

39 

42 

46 

49 

53 

56 

60 

64 

6 

8 

0 

3 

5 

8 

11 

14 

16 

19 

22 

25 

28 

31 

34 

37 

40 

43 

47 

50 

54 

57 

61 

8 

10 

0 

3 

5 

8 

10 

13 

15 

18 

21 

24 

26 

29 

32 

35 

38 

41 

44 

48 

51 

54 

58 

10 

11 

0 

2 

5 

7 

10 

13 

15 

18 

20 

23 

26 

28 

31 

34 

37 

40 

43 

46 

49 

53 

56 

11 

12 

0 

2 

5 

7 

10 

12 

15 

17 

20 

22 

25 

28 

30 

33 

36 

39 

42 

45 

48 

51 

54 

12 

13 

0 

2 

5 

7 

9 

12 

14 

17 

19 

22 

24 

27 

30 

32 

35 

38 

41 

44 

47 

50 

53 

13 

14 

0 

2 

5 

7 

9 

11 

14 

16 

19 

21 

24 

26 

29 

31 

34 

37 

40 

42 

45 

48 

51 

14 

15 

0 

2 

4 

7 

9 

11 

13 

16 

18 

20 

23 

25 

28 

30 

33 

36 

38 

41 

44 

47 

50 

15 

16 

0 

2 

4 

6 

9 

11 

13 

15 

18 

20 

22 

25 

27 

29 

32 

35 

37 

40 

43 

45 

48 

16 

17 

0 

2 

4 

6 

8 

10 

13 

15 

17 

19 

21 

24 

26 

29 

31 

33 

36 

39 

41 

44 

47 

17 

18 

0 

2 

4 

6 

8 

10 

12 

14 

16 

19 

21 

23 

25 

28 

30 

32 

35 

37 

40 

43 

45 

18 

19 

0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

27 

29 

31 

34 

36 

39 

41 

44 

19 

20 

0 

2 

4 

6 

8 

9 

11 

13 

15 

17 

19 

22 

24 

26 

28 

30 

33 

35 

37 

40 

42 

20 

21 

0 

2 

4 

5 

7 

9 

11 

13 

15 

17 

19 

21 

23 

25 

27 

29 

31 

34 

36 

38 

41 

21 

22 

0 

2 

3 

5 

7 

9 

11 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

35 

37 

39 

22 

23 

0 

2 

3 

5 

7 

8 

10 

12 

14 

15 

17 

19 

21 

23 

25 

27 

29 

31 

33 

36 

38 

23 

24 

0 

2 

3 

5 

6 

8 

10 

11 

13 

15 

17 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

24 

25 

0 

2 

3 

5 

6 

8 

9 

11 

13 

14 

16 

18 

19 

21 

23 

25 

27 

29 

31 

33 

35 

25 

26 

0 

3 

4 

6 

7 

9 

11 

12 

14 

15 

17 

19 

20 

22 

24 

26 

28 

29 

31 

33 

26 

27 

0 

3 

4 

6 

7 

9 

10 

12 

13 

15 

16 

18 

19 

21 

23 

25 

26 

38 

30 

32 

27 

28 

0 

3 

4 

5 

7 

8 

10 

11 

12 

14 

15 

17 

19 

20 

22 

23 

25 

27 

29 

30 

28 

29 

0 

3 

4 

5 

6 

8 

9 

11 

12 

13 

15 

16 

18 

19 

21 

22 

24 

26 

27 

29 

29 

30 

0 

2 

4 

5 

6 

7 

9 

10 

11 

13 

14 

15 

17 

18 

20 

21 

23 

24 

26 

28 

30 

31 

0 

2 

3 

5 

6 

7 

8 

9 

11 

12 

13 

15 

16 

17 

19 

20 

22 

23 

25 

26 

31 

32 

0 

2 

3 

4 

6 

7 

8 

9 

10 

11 

13 

14 

15 

16 

18 

19 

20 

22 

23 

25 

32 

33 

0 

2 

3 

4 

5 

6 

7 

8 

10 

11 

12 

13 

14 

15 

17 

18 

19 

21 

22 

23 

33 

34 

0 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

16 

17 

18 

19 

21 

22 

34 

35 

0 

2 

3 

4 

5 

6 

6 

7 

8 

9 

10 

11 

13 

14 

15 

16 

17 

IS 

19 

20 

35 

36 

0 

2 

3 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

36 

37 

0 

2 

2 

3 

4 

5 

6 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

37 

38 

0 

2 

3 

4 

4 

5 

6 

7 

7 

8 

9 

10 

11 

12 

13 

13 

14 

15 

16 

38 

39 

0 

2 

3 

3 

4 

5 

5 

6 

7 

8 

8 

9 

10 

11 

11 

12 

13 

14 

15 

39 

40 

0 

2 

2 

3 

4 

4 

5 

6 

6 

7 

fi 

8 

9 

10 

10 

11 

12 

13 

13 

40 

41 

0 

2 

2 

3 

3 

4 

4 

5 

6 

6 

£ 

7 

8 

9 

9 

10 

11 

11 

12 

41 

42 

0 

0 

1 

2 

2 

3 

3 

4 

4 

5 

5 

e 

7 

7 

8 

8 

9 

9 

10 

11 

42 

43 

0 

0 

1 

2 

2 

3 

3 

3 

4 

4 

5 

•je 

6 

6 

** 

7 

8 

8 

9 

9 

43 

44 

0 

0 

1 

1 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

6 

6 

7 

7 

8 

8 

44 

45 

0 

0 

1 

1 

1 

1 

2 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

6 

6 

6 

7 

45 

46 

0 

0 

0 

1 

1 

1 

1 

2 

2 

2 

2 

2 

3 

3 

4 

4 

4 

4 

5 

5 

5 

46 

47 

0 

0 

0 

1 

1 

1 

1 

1 

I 

2 

2 

2 

9 

2 

3 

3 

3 

3 

4 

4 

4 

47 

48 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

2 

2 

3 

48 

49 

0 

0 

0 

0 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

49 

50 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

50 

Add. 

\ 

30' 

II 
0 

3tf 

£ 

3# 

1 

30' 

i 

30 

6 

30 

6' 

30 

6' 

30" 

4 

3ft 

6' 

30 

* 

Add. 

Thermo  rn. 

0' 

1' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

10' 

Tficrrnoni. 

MEAN  REFRACTION. 

TABLE    IV.    B. 

Correction  of  the  Mean  Refraction  for  the  Height  of  the  Thermometer. 


Thermom. 

MEAN     REFRACTION. 

Thermom. 

O' 

1' 

2' 

3' 

4' 

5' 

6' 

V 

8' 

9' 

10' 

Subtract. 

4 

36 

\ 

30' 

4 

3ft 

6 

3ft 

4 

3ft 

6 

3ft 

6' 

3ft 

4 

30' 

6' 

3ft 

6' 

30 

6' 

subtract. 

o 
50 

•/ 

0 

0 

0 

50° 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

51 

0 

0 

0 

0 

0 

0 

0 

0 

0 

1 

i 

i 

i 

1 

1 

1 

i 

1 

i 

1 

i 

51 

52 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

i 

i 

i 

2 

2 

2 

2 

2 

2 

2 

3 

52 

53 

0 

0 

0 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

4 

4 

53 

54 

0 

0 

0 

1 

1 

1 

2 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

5 

5 

5 

54 

55 

0 

0 

1 

1 

1 

2 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

55 

56 

0 

0 

1 

1 

2 

2 

2 

3 

3 

4 

4 

4 

5 

5 

G 

6 

G 

7 

7 

8 

56 

57 

0 

0 

1 

2 

2 

2 

3 

3 

4 

4 

5 

5 

G 

6 

6 

7 

8 

8 

8 

9 

57 

58 

0 

0 

1 

2 

2 

3 

3 

4 

4 

5 

5 

6 

6 

7 

7 

8 

9 

9 

10 

10 

58 

59 

0 

1 

1 

2 

2 

3 

3 

4 

4 

5 

5 

6 

6 

7 

8 

8 

9 

10 

10 

11 

12 

59 

60 

0 

1 

1 

2 

2 

3 

3 

4 

5 

5 

6 

7 

7 

8 

9 

9 

10 

11 

11 

12 

13 

60 

61 

0 

1 

1 

2 

3 

3 

4 

4 

5 

6 

7 

7 

8 

9 

9 

10 

11 

12 

12 

13 

14 

61 

62 

0 

1 

1 

2 

3 

3 

4 

5 

6 

6 

7 

8 

9 

9 

10 

11 

12 

13 

14 

15 

15 

62 

63 

0 

1 

1 

2 

3 

4 

5 

5 

6 

7 

8 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

63 

64 

0 

1 

2 

2 

3 

4 

5 

6 

7 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

64 

65 

0 

1 

2 

3 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

65 

66 

0 

1 

2 

3 

4 

5 

6 

6 

7 

8 

9 

10 

11 

12 

14 

15 

16 

17 

18 

19 

20 

66 

67 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

16 

17 

18 

19 

20 

22 

67 

68 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

11 

11 

13 

14 

15 

1G 

18 

19 

20 

22 

23 

68 

69 

0 

1 

2 

3 

4 

5 

7 

8 

9 

10 

11 

12 

13 

15 

16 

17 

19 

20 

21 

23 

24 

69 

70 

0 

1 

2 

3 

5 

6 

7 

8 

9 

10 

12 

12 

14 

16 

17 

18 

20 

21 

22 

24 

25 

70 

71 

0 

1 

2 

4 

5 

6 

7 

8 

10 

11 

12 

13 

15 

16 

18 

19 

20 

22 

23 

25 

27 

71 

72 

0 

1 

2 

4 

5 

6 

8 

9 

10 

11 

13 

14 

16 

17 

18 

20 

21 

23 

25 

26 

28 

72 

73 

0 

1 

3 

4 

5 

7 

8 

9 

11 

12 

13 

14 

16 

18 

19 

21 

22 

24 

26 

27 

29 

73 

74 

0 

1 

3 

4 

5 

7 

8 

10 

11 

12 

14 

15 

17 

18 

20 

22 

23 

25 

27 

28 

30 

74 

75 

0 

1 

3 

4 

6 

7 

8 

10 

11 

13 

14 

16 

18 

19 

21 

22 

24 

26 

28 

29 

31 

75 

76 

0 

1 

3 

4 

6 

7 

9 

10 

12 

13 

15 

16 

18 

20 

22 

23 

25 

27 

29 

31 

32 

76 

77 

0 

1 

3 

5 

6 

8 

9 

11 

12 

14 

16 

17 

19 

21 

22 

24 

26 

28 

30 

32 

34 

77 

78 

0 

2 

3 

5 

6 

8 

9 

11 

13 

14 

16 

18 

20 

21 

23 

25 

27 

29 

31 

33 

35 

78 

79 

0 

2 

3 

5 

6 

8 

10 

11 

13 

15 

17 

18 

110 

22 

24 

26 

28 

30 

32 

34 

36 

79 

80 

0 

2 

3 

5 

7 

8 

10 

12 

14 

15 

17 

19 

21 

23 

25 

27 

29 

31 

83 

35 

37 

80 

81 

0 

2 

3 

5 

7 

9 

10 

12 

14 

16 

18 

20 

21 

24 

26 

28 

30 

32 

34 

36 

38 

81 

82 

0 

2 

5 

7 

9 

11 

13 

14 

16 

18 

20 

22 

24 

26 

28 

31 

33 

35 

37 

40 

82 

83 

0 

2 

5 

7 

9 

11 

13 

15 

17 

19 

21 

23 

25 

27 

29 

31 

34 

36 

38 

41 

83 

84 

0 

2 

6 

8 

9 

11 

13 

15 

17 

19 

21 

23 

26 

28 

30 

32 

35 

37 

39 

42 

84 

85 

0 

2 

6 

8 

10 

12 

14 

16 

18 

20 

22 

2426 

29 

31 

33 

36 

38 

40 

43 

85 

86 

0 

2 

6 

8 

10 

12 

14 

16 

18 

20 

23 

25  27 

29 

32 

34 

37 

39 

42 

44 

86 

87 

0 

2 

6 

8 

10 

12 

14 

17 

19 

21 

23 

25  28 

30 

32 

85 

38 

40 

43 

45 

87     . 

88 

0 

2 

4 

6 

8 

10 

13 

15 

17 

19 

21 

24 

26  28 

31 

33 

36 

38 

41 

44 

46 

88 

89 

0 

2 

4 

6 

9 

11 

13 

15 

17 

20 

22 

24 

27;  29 

32 

34 

37 

39 

42 

45 

48 

89 

90 

0 

2 

4 

7 

9 

11 

13 

16 

18 

20 

23 

25 

27  30 

32 

35 

38 

40 

43 

46 

49 

90 

91 

0 

2 

4 

7 

9 

11 

14 

16 

18 

21 

23 

25 

28|31 

33 

3G 

39 

41 

44 

47 

50 

91 

92 

0 

2 

5 

7 

9 

11 

14 

16 

19 

21 

24 

26 

29  31 

34 

37 

39 

42 

45 

48 

51 

92 

93 

0 

2 

5 

7 

9 

12 

14 

17 

19 

22 

24 

27 

29  32 

35 

37 

40 

43 

46 

49 

52 

93 

94 

0 

2 

5 

7 

10 

12 

14 

17 

19 

22 

25 

27 

3033 

35 

38 

41 

44 

47 

50 

53 

94 

95 

0 

2 

5 

7 

10 

12 

15 

17 

20 

22 

25 

28 

30!33 

3G 

39 

42 

45 

48 

51 

54 

95 

96 

0 

2 

5 

7 

10 

12 

15 

18 

20 

23 

26 

28 

31  34 

37 

40 

43 

46 

49 

52 

55 

96 

97 

0 

3 

5 

8 

10 

13 

15 

18 

21 

23 

26 

29 

32  35 

38 

41 

44  47 

50 

53 

56 

97 

98 

0 

3 

5 

8 

10 

13 

16 

18 

21 

24 

27 

29 

32  35 

38 

41 

44  48 

51 

54 

58 

98 

99 

0 

3 

5 

8 

11 

13 

16 

19 

21 

24 

27 

30 

33  36 

39 

42 

45  49 

52 

55 

59 

99 

100 

0 

3 

5 

8 

11 

13 

16 

19 

22 

25 

28 

31 

3437 

40 

43 

46  50 

53 

56 

60 

100 

Subtract. 

4 

3* 

1 

36 

0 

3ft 

0 

3ft 

4 

30 

4 

30 

&   3ft 

4 

3ft 

0   30 

6' 

3ft 

6 

Subtract. 

JT  hcrrnom. 

0 

1' 

2' 

3' 

4' 

5' 

6' 

7' 

§' 

9' 

10' 

Thflrmnm 

MEAN     REFRACTION. 

x  nemioin  > 

23 


TABLE    V.    LOG.    A. 

Logs  A,  B,  C,  and  D,  for  Computing  the  First  Correction  of  the  lunar  L/istanco. 


App. 

Alt 

REDUCED  PARALLAX  AND  REFRACTION  O.F  D. 

Alt. 

of  J> 

41' 

42' 

43' 

44' 

45' 

46' 

47' 

4S' 

49' 

5O' 

51 

5*2' 

S3' 

54' 

5ff 

5°  0' 

.0288 

0295 

0301 

0308 

0315 

0321 

0328 

0335 

0341 

0348 

0355 

0361 

036& 

2 

.0286 

0293 

0299 

0306 

0313 

0319 

0326 

0333  0339 

0346 

0352 

0359 

0366 

4 

.0284 

0291 

0297 

0304 

0311 

0317 

0324 

0330  0337 

0344 

0350 

0357 

0363 

! 
1 

6 

.0282 

0289 

0296 

0302 

0309 

0315 

0322 

0328 

0335 

0341 

0348 

0354 

0361 

8 

.0281 

0287 

0294 

0300 

0307 

0313 

0320 

0326 

0333 

0339 

0346 

0352 

0359 

5  10 

.0279 

0285 

0292 

0298 

0305 

0311 

0318 

0324 

0331 

0337 

0344 

0350 

0356 

12 

.0277 

0284 

0290 

0296 

0303 

0309 

0316 

0322 

0329 

0335 

0341 

0348 

0354 

14 

.0275 

0282 

0288 

0295 

0301 

0307 

0314 

0320 

0327 

0333 

0339 

0346 

0352 

16 

.0274 

0280 

0286 

0293 

0299 

0306 

0312 

0318 

0325 

0331 

0337 

0344 

0350 

18 

.0272 

0278 

0285 

0291 

0297 

0304 

0310 

0316 

0323 

0329 

0335 

0341 

0348 

5  20 

.0270 

0277 

0283 

0289 

0296 

0302 

0308 

0314 

0321 

0327 

0333 

0339 

0346 

22 

.0269 

0275 

0281 

0288 

0294 

0300 

0306 

0313 

0319 

0325 

0331 

0337 

0344 

24 

.0267 

0273 

0280 

0286 

0292 

0298 

0304 

0311 

0317 

0323 

0329 

0335 

0341 

26 

.0265 

0272 

0278 

0284 

0290 

0296 

0303 

0309 

0315 

0321. 

0327 

0333 

0339 

0346  { 

28 

.0264 

0270 

0276 

0282 

0289 

0295 

0301 

0307 

0313 

0319 

0325 

0331 

0337 

0344 

5  30 

.0262 

0268 

0275 

0281 

0287 

0293 

0299 

0305 

0311 

0317 

0323 

0329 

0335 

0342 

32 

.0261 

0267 

0273 

0279 

0285 

0291 

0297 

0303 

0309 

0315 

0321 

0327 

0334 

0340 

34 

.0259 

0265 

0271 

0277 

0283 

0290 

0296 

0302 

0308 

0314 

0320 

0326 

0332 

0338 

36 

.0258 

0264  j  0270 

0276 

0282 

0288 

0294 

0300 

0306 

0312 

0318 

0324 

0330 

'J336 

38 

0262 

0268 

0274 

0280 

0286 

0292 

0298 

0304 

0310 

0316 

0322 

0328 

033-1 

1 

5  40 

0261 

0267 

0273 

0279 

0285 

0290 

0296 

0302 

0308 

0314 

0320 

03S6 

0?,32 

1 

42 

0259 

0265 

0271 

0277 

0283 

0289 

0295 

0301 

0306 

0312 

0318 

Of,24 

0330 

44 

0258!  0264 

0270 

0275 

0281 

0287 

0293 

0299 

0305 

0311 

0316 

0322 

0328 

46 

0256 

0262 

0268 

0274 

0280 

0286 

0291 

0297 

0303 

0309 

0315 

0320 

0326 

48 

0255 

0261 

0267 

0272 

0278 

0284 

0290 

0296 

0301 

0307 

0313 

0319 

0324 

5  50 

0253 

0259 

0265 

0271 

0277 

0282 

0288 

0294 

0300 

0305 

0311 

0317 

0323 

'  52 

0252 

0258 

0264 

0269 

0275 

0281 

0287 

0292 

0298 

0304 

0309 

0315 

0321 

54 

0251 

0256 

0262 

0268 

0274 

0279 

0280 

0291 

0296 

0302 

0308 

0313 

0319 

56 

0249 

0255 

0261 

0266 

0272 

0278 

0283 

0289 

0295 

0300 

0306 

0312 

0817 

58 

0248 

0254 

0259 

0265 

0271 

0276 

0282 

0287 

0293 

0299 

0304 

0310 

0316 

6  0 

0247 

0252 

0258 

0263 

0269 

0275 

0280 

0286 

0291 

0297 

0303 

0308 

0314 

2 

0245 

0251 

0256 

0262 

0268 

0273 

0279 

0284 

0290 

0295 

0301 

0307 

0312 

4 

0244 

0249 

0255 

0261 

0266 

0272 

0277 

0283 

0288 

0294 

0299 

0305 

0310 

6 

0243 

0248 

0254 

0259 

0265 

0270 

0276 

0281 

0287 

0292 

0298 

0303 

0309 

8 

0241 

0247 

0252 

0258 

0263 

026'J 

0274 

0280 

0285 

0291 

0296 

0302 

0307 

, 

6  10 

0240 

0246 

0251 

0256 

0262 

0267 

0273 

0278 

0284 

0289 

0295 

0300 

0306 

12 

0239 

0244 

0250 

0255 

0261 

0266 

0271 

0277 

0282 

0288 

0293 

0299 

0304 

14 

0237 

0243 

0248 

0254 

0259 

0265 

0270 

0275 

0281 

.0286 

0292 

0297 

0302 

16 

0236 

0242 

0247 

0252 

0258 

0263 

0269 

0274 

0279 

0285 

0290 

0295 

0301 

18 

0235 

0240 

0246 

0251 

0257 

0262 

0267 

0273 

0278 

0283 

0289 

0294 

0299 

6  20 

0234 

0239 

0245 

0250 

0255 

0261 

0266 

0271 

0276 

0282 

0287 

0292 

0298 

22 

0233 

0238 

0243 

0249 

0254 

0259 

0264 

0270 

0275 

0280 

0286 

0291 

0296 

24 

0231 

0237 

0242 

0247 

0253 

0258 

0263 

0268 

0274 

0279 

0284 

0289 

0295 

26 

0236 

0241 

0246 

0251 

0257 

0262 

0267 

0272 

0277 

0283 

0288 

0293 

28 

0234 

0240 

0245 

0250 

0255 

0260 

0266 

0271 

0276 

0281 

0286 

0292 

0297 

6  30 

0233 

0238 

0244 

0249 

0254 

0259 

0264 

0270 

0275 

0280 

0285 

0290 

0295 

32 

0232 

0237 

0242 

0248 

0253 

0258 

0263 

0268 

0273 

0278 

0284 

0289 

0294 

34 

0231 

0236 

0241 

0246 

0251 

0257 

0262 

0267 

0272 

0277 

0282 

0287 

0292 

36 

0230 

0235 

0240 

0245 

0250 

0255 

0260 

0266 

0271 

0276 

0281 

0286 

0291 

38 

0229 

0234 

0239 

0244 

0249 

0254 

0259 

0264 

0269 

0274 

0279 

0284 

0290 

6  40 

0227 

0232 

0238 

0243 

0248 

0253 

0258 

0263 

0268 

0273 

0278 

0283 

0288 

42 

0220 

0231 

0236 

0241 

0246 

0252 

0257 

0262 

0267 

0272 

0277 

0282 

0287 

44 

02?.5 

0230 

0235 

0240 

0245 

0250 

0255 

0260 

0265 

0270 

0275 

0280 

0285 

46 

0224 

0229 

0234 

0239 

0244 

0249 

0254 

0259 

0264 

0269 

0274 

0279 

0284 

48 

0223 

0228 

0233 

0238 

0243 

0248 

0253 

0258 

0263 

0268 

0273 

0278 

0283 

6  50 

0222 

0227 

0232 

0237 

0242 

0247 

0252 

0257 

0262 

0266 

0271 

0276 

0281 

52 

0221 

0226 

0231 

0236 

0241 

0246 

0250 

0255 

0260 

0265 

0270 

0275 

0280 

54 

0220 

0225 

0230 

0235 

0239 

0244 

0249 

0254 

0259 

0264 

0269 

0274 

0279 

56 

0219 

0224 

0229  0233 

0238 

C243 

0248 

0253 

0258 

0263 

0267 

0272 

0277 

58 

0218 

0223 

0227 

0232  0237 

0242 

0247 

0252 

0257 

0261 

0266 

0271 

0276 

7  0 

0217" 

0722 

0226 

0231  0236 

0241 

0246 

0251 

0255 

0260 

0265 

0270 

0275 

24 


TABLE    V.    LOG.  A. 

•  Logs.  A,  B,  C,  and  D,  for  Lunars. 


AA?f 

REDUCED  PARALLAX  AND  REFRACTION  OF  d. 

ot  }>. 

44' 

45' 

46' 

47' 

4§' 

49' 

50' 

51' 

52' 

53' 

54' 

55' 

56' 

57' 

7°  0' 

.0222 

0226 

0231 

0236 

0241 

0246 

0251 

0255 

0260 

0265 

0270 

0275 

3 

.0220 

0225 

0230 

0234 

0239 

0244 

0249 

0254 

0258 

0263 

0268 

0273 

6 

.0218 

0223 

0228 

0233 

0238 

0242 

0247 

0252 

0257 

0261 

0266 

0271 

9 

.0217 

0222 

0226 

0231 

0236 

0241 

0245 

0250 

0255 

0260 

0264 

0269 

12 

.0215 

0220 

0225 

0230 

0234 

0239 

0244 

0248 

0253 

0258 

0262 

0267 

7  15 

.0214 

0219 

0223 

0228 

0233 

0237 

0242 

0247 

0251 

0256 

0261 

0265 

18 

.0213 

0217 

0222 

0226 

0231 

0236 

0240 

0245 

0250 

0254 

0259 

0263 

21 

.0211 

0216 

0220 

0225 

0230 

0234 

0239 

0243 

0248 

0253 

0257 

0262 

24 

.0210 

0214 

0219 

0223 

0228 

0233 

0237 

0242 

0246 

0251 

0255 

0260 

27 

.0208 

0213 

0217 

0222 

0227 

0231 

0236 

0240 

0245 

0249 

0254 

0258 

7  30 

.0207 

0211 

0216 

0220 

0225 

0230 

0234 

0239 

0243 

0248 

0252 

0257 

33 

.0206 

0210 

0215 

0219 

0224 

0228 

0232 

0237 

0241 

0246 

0250 

0255 

36 

.0204 

0209 

0213 

0218 

0222 

0227 

0231 

0235 

0240 

0244 

0249 

0253 

39 

.0203 

0207 

0212 

0216 

0221 

0225 

0229 

0234 

0238 

0243 

0247 

0252 

42 

.0202 

0206 

0210 

0215 

0219 

0224 

0228 

0232 

0237 

0241 

0246 

0250 

7  45 

.0200 

0205 

0209 

0213 

0218 

0222 

0227 

0231 

0235 

0240 

0244 

0248 

48 

.0199 

0203 

0208 

0212 

0216 

0221 

0225 

0229 

0234 

0238 

0242 

0247 

51 

.0198 

0202 

0206 

0211 

0215 

0219 

0224 

0228 

0232 

0237 

0241 

0245 

0249 

54 

.0136 

0201 

0205 

0209 

0214 

0218 

0222 

0227 

0231 

0235 

0239 

0244 

0248 

57 

.0195 

0200 

0204 

0208 

0212 

0217 

0221 

0225 

0229 

0234 

0238 

0242 

0246 

8  0 

.0194 

0198 

0203 

0207 

0211 

0215 

0219 

0224 

0228 

0232 

0236 

0241 

0245 

• 

3 

.0193 

0197 

0201 

0206 

0210 

0214 

0218 

0222 

0227 

0231 

0235 

0239 

0243 

6 

.0192 

0196 

0200 

0204 

0208 

0213 

0217 

0221 

0225 

0229 

0233 

0238 

0242 

9 

0195 

0199 

0203 

0207 

0211 

0215 

0220 

0224 

0228 

0232 

0236 

0240 

12 

0193 

0198 

0202 

0206 

0210 

0214 

0218 

0222 

0227 

0231 

0235 

0239 

8  15 

0192 

0196 

0201 

0205 

0209 

0213 

0217 

0221 

0225 

0229 

0233 

0237 

18 

0191 

0195 

0199 

0203 

0207 

0212 

0217 

0220 

0224 

0228 

0232 

0236 

21 

0190 

0194 

0198 

0202 

0206 

0210 

0214 

0218 

0222 

0226 

0231 

0235 

24 

0189 

0193 

0197 

0201 

0205 

0209 

0213 

0217 

0221 

0225 

0229 

0233 

27 

0188 

0192 

0196 

0200 

0204 

0208 

0212 

0216 

0220 

0224 

0228 

0232 

8  30 

0187 

0191 

0195 

0199 

0203 

0207 

0211 

0215 

0219 

0223 

0226 

0230 

33 

0186 

0190 

0193 

0197 

0201 

0205 

0209 

0213 

0217 

0221 

0225 

0229 

36 

0184 

0188 

0192 

0196 

0200 

0204 

0208 

0212 

0216 

0220 

0224 

0228 

39 

0183 

OJ87 

0191 

0195 

0199 

0203 

0207 

0211 

0215 

0219 

0223 

0226 

42 

0182 

0186 

0190 

0194 

0198 

0202 

0206 

0210 

0214 

0217 

0221 

0225 

8  45 

0181 

0185 

0189 

0193 

0197 

0201 

0205 

0208 

0212 

0216 

0220 

0224 

48 

0180 

0184 

0188 

0192 

0196 

0200 

0203 

0207 

0211 

0215 

0219 

0223 

51 

0179 

0183 

0187 

0191 

0195 

0198 

0202 

0206 

0210 

0214 

0218 

0221 

54 

0178 

0182 

0186 

0190 

0193 

0197. 

0201 

0205 

0209 

0212 

0216 

0220 

57 

0177 

0181 

0185 

0189 

0192 

0196 

0200 

0204 

0208 

0211 

0215 

0219 

9  0 

0176 

0180 

0184 

0188 

0191 

0195 

0199 

0203 

0206 

0210 

0214 

0218 

3 

0175 

0179 

0183 

0186 

0190 

0194 

0198 

0201 

0205 

0209 

0213 

0216 

6 

0174 

0178 

0182 

0185 

0189 

0193 

0197 

0200 

0204 

0208 

0211 

0215 

9 

0173 

0177 

0181 

0184 

0188 

0192 

0196 

0199 

0203 

0207 

0210 

0214 

12 

0172 

0176 

0180 

0183 

0187 

0191 

0194 

0198 

0202 

0206 

0209 

0213 

9  15 

0171 

0175 

0179 

0182 

0186 

0190 

0193 

0197 

0201 

0204 

0208 

0212 

18 

0170 

0174 

0178 

0181 

0185 

0189 

0192 

0196 

0200 

0203 

0207 

0211 

21 

0170 

0173 

0177 

0180 

0184 

0188 

0191 

0195 

0199 

0202 

0206 

0209 

24 

0172 

0176 

0179 

0183 

0187 

0190 

0194 

0198 

0201 

0205 

0208 

27 

0171 

0175 

0179 

0182 

0186 

0189 

0193 

0196 

0200 

0204 

U207 

9  30 

0170 

0174 

0178 

0181 

0185 

0188 

0192 

0195 

0199 

0203 

0206 

33 

0170 

0173 

0177 

0180 

0184 

0187 

0191 

0194 

0198 

0201 

0205 

36 

0169 

0172 

0176 

0179 

0183 

0186 

0190 

0193 

0197 

0200 

0204 

39 

0168 

0171 

0175 

0178 

0182 

0185 

0189 

0192 

0196 

0199 

0203 

42 

0167 

0170 

0174 

0177 

0181 

0184 

0188 

0191 

0195 

0198 

0202 

9  45 

0166 

0169 

0173 

0176 

0180 

0183 

0187 

0190 

0194 

0197 

0201 

48 

0165 

0169 

0172 

0176 

0179 

0182 

0186 

0189 

0193 

0196 

0200 

0203 

51 

0164 

0168 

0171 

0175 

0178 

0182 

0185 

0188 

0192 

0195 

0199 

0202 

54 

0163 

0167 

0170 

0174 

0177 

0181 

0184 

0187 

0191 

0194 

0198 

0201 

57 

0163 

0166 

0169 

0173 

0176 

0180 

0183 

0186 

0190 

0193 

0197 

0200 

10  0 

0162 

0165 

0169 

0172 

0175 

0179 

0182 

0186 

0189 

0192 

0196 

0199 

25 


TABLE    V.    LOG.  A, 

Logs.  A,  B,  C,  and  D,  for  Lunars. 


ir 

of  >' 

REDUCED  PARALLAX  AND  REFRACTION  OF  > 

46' 

49" 

48' 

49' 

50' 

51' 

52' 

53' 

54' 

55' 

56' 

57' 

58' 

10°0' 

•0162 

0165 

0169 

0172 

0175 

0179 

0182 

0186 

0189 

0192 

0196 

0199 

5 

•0160 

0164 

0167 

0171 

0174 

0177 

0181 

0184 

0187 

0191 

0194 

0197 

10 

•0159 

0162 

0166 

0169 

0172 

0176 

0179 

0182 

0186 

0189 

0192 

0196 

15 

•0158 

0161 

0164 

0168 

0171 

0174 

0178 

0181 

0184 

0187 

0191 

0194 

20 

•0156 

0160 

0163 

0166 

0170 

0173 

0176 

0179 

0183 

0186 

0189 

0192 

25 

•0155 

0158 

0162 

0165 

0168 

0171 

0175 

0178 

0181 

0184 

0188 

0191 

1030 

.0154 

0157 

0160 

0164 

0167 

0170 

0173 

0177 

0180 

0183 

0186 

0189 

35 

.0153 

0156 

0159 

0162 

0166 

0169 

0172 

0175 

0178 

0181 

0185 

0188 

40 

.0151 

0155 

0158 

0161 

0164 

0167 

0171 

0174 

0177 

0180 

0183 

0186 

45 

.0150 

0153 

0157 

0160 

0163 

0166 

0169 

0172 

0175 

0179 

0182 

0185 

50 

.0149 

0152 

0155 

0158 

0162 

0165 

0168 

0171 

0174 

0177 

0180 

0183 

55 

.0148 

0151 

0154 

0157 

0160 

0163 

0167 

0170 

0173 

0176 

0179 

0182 

11  0 

.0147 

0150 

0153 

0156 

0159 

0162 

0165 

0168 

0171 

0174 

0177 

0181 

5 

.0146 

0149 

0152 

0155 

0158 

0161 

0164 

0167 

0170 

0173 

0176 

0179 

10 

0148 

0151 

0154 

0157 

0160 

0163 

0166 

0169 

0172 

0175 

0178 

15 

0146 

0149 

0152 

0155 

0158 

0161 

0164 

0167 

0170 

0173 

0176 

20 

0145 

0148 

0151 

0154 

0157 

0160 

0163 

0166 

0169 

0172 

0175 

25 

0144 

0147 

0150 

0153 

0156 

0159 

0162 

0165 

0168 

0171 

0174 

1130 

0143 

0146 

0149 

0152 

0155 

0158 

0161 

0164 

0167 

0170 

0172 

35 

0142 

0145 

0148 

0151 

0154 

0157 

0160 

0162 

0165 

0168 

0171 

40 

0141 

0144 

0147 

0150 

0153 

0156 

0158 

0161 

0164 

0167 

0170 

45 

0140 

0143 

0146 

0149 

0151 

0154 

0157 

0160 

0163 

0166 

0169 

50 

0139 

0142 

0145 

0148 

0150 

0153 

0156 

0159 

0162 

0165 

0167 

55 

0138 

0141 

0144 

0146 

0149 

0152 

0155 

0158 

0161 

0163 

0166 

12  0 

0137 

0140 

0143 

0145 

0148 

0151 

0154 

0157 

0159 

0162 

0165 

5 

0136 

0139 

0142 

0144 

0147 

0150 

0153 

0156 

0158 

0161 

0164 

10 

0135 

0138 

0141 

0143 

0146 

0149 

0152 

0154 

0157 

0160 

0163 

15 

0134 

0137 

0140 

0142 

0145 

0148 

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0153 

0156 

0159 

0162 

20 

0133 

0136 

0139 

0141 

0144 

0147 

0150 

0152 

0155 

0158 

0160 

25 

0132 

0135 

0138 

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0143 

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0148 

0151 

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0157 

0159 

1230 

0131 

0134 

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0147 

0150 

0153 

0155 

0158 

35 

0130 

0133 

0136 

0138 

0141 

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0146 

0149 

0152 

0154 

0157 

40 

0129 

0132 

0135 

0137 

0140 

0143 

0145 

0148 

0151 

0153 

0156 

45 

0129 

0131 

0134 

0136 

0139 

0142 

0144 

0147 

0150 

0152 

0155 

0158 

50 

0128 

0130 

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0138 

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0146 

0149 

0151 

0154 

0156 

55 

0127 

0129 

0132 

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0140 

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0145 

0148 

0150 

0153 

0155 

13  0 

0126 

0129 

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0134 

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0144 

0147  0149 

0152 

0154 

5 

0125 

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10 

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0123 

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0150 

25 

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0127 

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0132 

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0139 

0142 

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0149 

1330 

0121 

0124 

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0131 

0133 

0136 

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0141 

0143 

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0148 

35 

0120 

0123 

0125 

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0130 

0133 

0135 

0138 

0140 

0142 

0145 

0147 

40 

0120 

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0124 

0127 

0129 

0132 

0134 

0137 

0139 

0142 

0144 

0146 

45 

0121 

0124 

0126 

0128 

0131 

0133 

0136 

0138 

0141 

0143 

0145 

50 

0120 

0123 

0125 

0128 

0130 

0132 

0135 

0137 

0140 

0142 

0145 

55 

0120 

0122 

0124 

0127 

0129 

0132 

0134 

0136 

0139  0141 

0144 

14  0 

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0124 

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0138  0140 

0143 

5 

0118 

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0135 

0137 

0139 

0142 

10 

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0124 

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0129 

0132 

0134 

0136 

0139 

0141 

15 

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0119 

0121 

0124 

0126 

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0131 

0133 

0135 

0138 

0140 

20 

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0118 

0121 

0123 

0125 

0128 

0130 

0132 

0135 

0187 

0139 

25 

0115 

0118 

0120 

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0124 

0127 

0129 

0131 

0134 

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0138 

1430 

0114 

0117 

0119 

0121 

0124 

0126 

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0131 

0133 

0135 

0137 

35 

0114 

0116 

0118 

0121 

0123 

0125 

0128 

0130 

0132 

0134 

0137 

40 

0113 

0115 

0118 

0120 

0122 

0124 

0127 

0129 

0131 

0134 

0136 

45 

0112 

0115 

0117 

0119 

0121 

0124 

0126 

0128 

0130 

0133 

0135 

50 

0112 

0114 

0116 

0118 

0121 

0123 

0125 

0127 

0130 

0132 

0134 

55 

0111 

0113 

0116 

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0124 

0127 

0129 

0131 

0133 

15  0 

0110 

0113 

0115 

0117 

0119 

0121 

0124 

0126 

0128 

0130 

0133 

26 


TABLE    V.    LOG.  A. 

Logs.  A,  B,  C,  and  D,  for  Lunars. 


APP 

REDUCED  PARALLAX  AND  REFRACTION  OF  ]>. 

Alt. 
of  )>. 

18 

19 

5O' 

51' 

52' 

53 

54' 

55' 

56' 

57' 

58' 

59' 

15°  0 

.0110 

0113 

0115 

0117 

0119 

0121 

0124 

0126 

0128 

0130 

0133 

10 

.0109 

0111 

0113 

0116 

0118 

0120 

0122 

0124 

0127 

0129 

0131 

20 

.0108 

0110 

0112 

0114 

0116 

0119 

0121 

0123 

0125 

0127 

0129 

I    30 

.0107 

0109 

0111 

0113 

0115 

0117 

0119 

0121 

0124 

0126 

0128 

,    40 

.0105 

0107 

0110 

0112 

0114 

0116 

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0122 

0124 

0126 

50 

.0104 

0106 

0108 

0110 

0112 

0115 

0117 

0119 

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16  0 

.0103 

0105 

0107 

0109 

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0113 

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0119 

0121 

0124 

10 

.0102 

0104 

0106 

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0116 

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20 

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0106 

0108 

0109 

18  0 

0092 

0094 

0096 

0098 

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0101 

0103 

0105 

0107 

0108 

10 

0091 

0093 

0095 

0097 

0098 

0100 

0102 

0104 

0105 

0107 

0109 

20 

0090 

0092 

0094 

0096 

0097 

0099 

0101 

0103 

0104 

0106 

0108 

30 

0089 

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0102 

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40 

0088 

0090 

0092 

0094 

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0104 

0106 

50 

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0096 

0098 

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19  0 

0087 

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0102 

0104 

10 

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0101 

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20 

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30 

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20  0 

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10 

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20 

0080 

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50 

0078 

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009JL 

0093 

21  0 

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40 

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40 

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50 

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0069 

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0075 

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0079 

24  0 

0067 

0068 

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0072 

0073 

0074 

0076 

0077 

0078 

10 

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0067 

0069 

0070 

0071 

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0074 

0075 

0076 

0078 

20 

0066 

0067 

0068 

0069 

0071 

0072 

0073 

0074 

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30 

0065 

0066 

0068 

0069 

0070 

0071 

0072 

0074 

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40 

0065 

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0067 

0068 

0069 

0071 

0072 

0073 

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0076 

50 

0064 

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0068 

0069 

0070 

0071 

0072 

0074 

0075 

25  0 

0063 

0065 

0066 

0067 

0068 

0069 

0071 

0072 

0073 

0074 

TABLE    V.    LOG.  A, 

Logs.  A,  B,  C,  and  D,  for  Lunars. 


ll? 

REDUCED  PARALLAX  AND  REFRACTION  OF  > 

All. 

of  D- 

50' 

51' 

52' 

53' 

54' 

55' 

56' 

57' 

58' 

59' 

oo 

25°  0' 

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0065 

0066 

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0069 

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40 

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26  0 

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0064 

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28  0 

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20 

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20 

0044 

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40 

0044 

0045  !  0045 

0046 

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0051 

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33  0 

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20 

.0042 

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40 

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34  0 

.0041 

0042  i  0043 

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0045 

0046 

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20 

.0040 

0041  j  0042 

0043 

0043 

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0046 

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40 

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'0041 

0041 

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0045 

0046 

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35  0 

.0039 

0040 

0041 

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0047 

20 

.0039  1  0039  0040 

0041 

0042 

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0046 

40 

.0038  0039  !  0039 

0040 

0041 

0042 

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0045 

36  0 

.0037 

0038  |  0039 

0040 

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0042 

0043 

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0044 

20 

.0037 

0038,  0038 

0039 

0040 

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0042 

0042 

0043 

0044 

40 

.0036 

0037 

0038 

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0041 

0042 

0042 

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37  0 

.0036 

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20 

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38  0 

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20 

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40  0 

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20 

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20 

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0033  0033 

0034 

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40 

0029 

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0033 

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42  0 

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0030 

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0034 

20 

0029 

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0031  0032 

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0033 

40 

0028 

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0032 

0032 

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43  0 

0028 

0028 

0029 

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0030 

0030  0031 

0031 

0032 

0032 

20 

0027 

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0031 

0032 

40 

0027 

0027 

0028 

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0029 

0029 

0030 

0030 

0031  0031 

44  0 

0026 

0027 

0027 

0028 

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0030 

0030  0031 

20 

0026 

0026 

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0027 

0028 

0028 

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0030 

40 

0026 

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0028 

0028 

0029 

0029 

0030 

45  0 

0025  0026 

0026 

0027 

0027 

0027  0028 

0028 

0029 

0029 

TABLE    V.    LOG.  A. 

Logs.  A,  B,  C,  and  D,  for  Lunars. 


l?tp- 

REDUCED  PARALLAX  AND  REFRACTION  OF  3>.            j 

of  >« 

51 

52 

53' 

54' 

55' 

56' 

57' 

58' 

59' 

6O' 

! 

45°  0' 

.0025 

0026 

0026 

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46  0 

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0015 

0015 

0016 

0016 

0016 

0016 

57  0 

.0014 

0014 

0014 

0015 

0015 

0015 

0015 

0015 

0016 

0016 

30 

.0014 

0014 

0014 

0014 

0014 

0015 

0015 

0015 

0015 

0015 

58  0 

.0013 

0013 

0014 

0014 

0014 

0014 

0014 

0015 

0015 

0015 

30 

.0013 

0013 

0013 

0013 

0014 

0014 

0014 

0014 

0014 

0015 

59  0 

.0012 

0013 

0013 

0013 

0013 

0013 

0014 

0014 

0014 

0014 

30 

.0012 

0012 

0012 

0013 

0013 

0013 

0013 

0013 

0014 

0014 

60 

.0012 

0012 

0012 

0012 

0013 

0013 

0013 

0013 

0013 

0013 

61 

.0011 

0011 

0011 

0012 

0012 

0012 

0012 

0012 

0012 

0013 

62 

.0011 

0011 

0011 

0011 

0011 

0011 

0011 

0012 

0012 

0012 

63 

.0010 

0010 

0010 

0010 

0011 

0011 

0011 

0011 

0011 

0011 

64 

.0009 

0010 

0010 

0010 

0010 

0010 

0010 

OOlc 

0010 

0011 

65 

.0009 

0009 

0009 

0009 

0009 

0009 

0010 

0010 

0010 

0010 

66 

.0008 

0008 

0009 

0009 

0009 

0009 

0009 

0009 

0009 

0009 

67 

.0008 

0008 

0008 

0008 

0008 

0008 

0008 

0009 

0009 

0009 

68 

.0007 

0007 

0008 

0008 

0008 

0008 

0008 

0008 

0008 

0008 

69 

.0007 

0007 

0007 

0007 

0007 

0007 

0007 

0008 

0008 

0008 

70 

.0007 

0007 

0007 

0007 

0007 

0007 

0007 

0007 

0007 

0007 

71 

.0006 

0006 

0006 

0006 

0006 

0006 

0007 

0007 

0007 

0007 

72 

.0006 

0006 

0006 

0006 

0006 

0006 

0006 

0008 

0006 

0006 

73 

.0005 

0005 

0006 

0006 

0006 

0006 

0006 

0006 

0006 

0006 

74 

.0005 

0005 

0005 

0005 

0005 

0005 

0005 

0005 

0005 

0006 

75 

.0005 

0005 

0005 

0005 

0005 

0005 

0005 

0005 

0005 

0005 

76 

.0004 

0005 

0005 

0005 

0005 

0005 

0005 

0005 

0005 

0005 

77 

.0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

78 

.0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

79 

.0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

80 

.0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

0004 

81 

.0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

82 

.0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

83 

.0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

84 

.0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

85 

.0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

1 

I 

86 

.0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

87 

.0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

• 

1 

88 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

89 

.0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

!)0 

.0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

0003 

TABLE    V.    LOG.    B. 

Logs.  A,  B,  C,  and  D,  for  Lunars. 


||   App.  Alt. 

nf 

REDUCED  REFRACTION  AND  PARALLAX  OF  ©  OR  -)f. 

OI 

Oor*. 

O'O" 

O'iO" 

I'O" 

I'M" 

2'0" 

r  so" 

8'0" 

I1  80" 

4/0" 

4'  10" 

S'O" 

5'  SO" 

5°     0' 

10 

20 

30 

40 

50 

6       0 

20 

9.9970 

40 

9.9972 

7       0 

9.9976 

9.9974 

20 

9.9977 

9.9975 

40 

9.9981 

9.9978 

9.9976 

8       0 

9.9982 

9.9979 

9.9977  ' 

20 

9.9982 

9.9980 

9.9978 

40 

9.9983 

9.9981 

9.9979 

9       0 

9.9986 

9.9984 

9.9982 

9.9980 

20 

9.9986 

9.9985 

9.9983 

9.9981 

40 

9.9987 

9.9985 

9.9983 

9.9982 

10      0 

9.9989 

9.9988 

9.9986 

9.9984 

9.9982 

11 

9.9992 

9.9991 

9.9989 

9.9987 

9.9986 

9.9984 

12 

9.9993 

9.9992 

9.9990 

9.9989 

9.9987 

9.9986 

13 

9.9995 

9.9994 

9.9992 

9.9991 

9.9990 

9.9989 

9.9987 

14 

9.9995 

9.9994 

9.9993 

9.9992 

9.9991 

9.9990 

15 

9.9997 

9.9996 

9.9995 

9.9994 

9.9993 

9.9992 

9.9991 

16 

9.9997 

9.9996 

9.9995 

9.9994 

9.9993 

9.9993 

18 

9.9990 

9.9998 

9.9997 

9.9996 

9.9995 

9.9995 

20 

0.0000 

9.9999  9.999S 

9.9998 

9.9997 

9.9996 

9.9996 

25 

0.0000 

0.0000  9.9999 

9.9999 

9.9998 

9.9998 

30 

00001 

0.0001 

0.0000  0.0000 

0.0000 

9.9999 

50 

0.0001 

0.0001 

0.0001 

0.0001 

0.0001 

0.0001 

90 

0.0001  0.0002 

0.0002 

0.0002 

LOG.    C. 

App.  AH. 

REDUCED  REFRACTION  AND  PARALLAX  OF  Q  OR  •#. 

of 
Oor*. 

O'O" 

0'  30" 

I'O" 

I'M" 

I'D" 

I1  10" 

S'O" 

3'  SO" 

4/0" 

4'  30" 

§'  0" 

5'  10" 

5°      0' 

20 

40 

6       0 

20 

9.9969 

40 

9.9970 

7       0 

9.9974 

9.9972 

8 

9.9980 

9.9978  9.9975 

9 

9.9984 

9.9982  9.9980 

9.9978 

10 

9.9988 

9.9986 

9.9984  9.9982 

9.9981 

11 

9.9990 

9.9989  9.9987 

9.9986  9.9984 

9.9982 

12 

9.9991  9.9990:9.9988 

9.9987  9.9985 

9.9984 

13 

9.9993 

9.9992  9.9991  9.9989 

9.998819.9987 

9.9985 

14 

9.9994 

9.9993  9.9991  9.9990 

9.9989  9.9988 

15 

9.9995 

9.9994 

9.9993 

9.9992  9.9991 

9.9990 

9.9989 

16 

99996 

9.9995 

9.9994 

9.9993  9.9992 

9.9990 

17 

9.9996 

9.9995 

9.9994 

9.9993  9.9992 

9.9991 

i 

18 

9.9997  9.9996 

9.9995 

9.9994 

9.9994  9.9993 

20 

9.9998 

9.9998  9.9997 

9.9996 

9.9995 

9.9994  9.9993 

25 

9.9999 

9.9998  9.9998 

9.9997 

99996 

9.9996 

30 

0.0000 

9.9999 

9.9999  9.9998 

9.9998 

9.9997 

40 

0.0000  9.9999 

9.9999  99999 

9.9999 

,    50 

0.0000  0.0000  0.0000 

9.9999  9.9999 

«L 

o.jooo  o.oooo  o.oooo  ;  o.oooo 

TABLE    V.    LOG.    B, 

Logs.  A,  B,  C,  and  D,  for  Lunars. 


App.  Alt. 

_^« 

REDUCED  REFRACTION  AND  PARALLAX  OF  ©  OR  ^-. 

ot 

6'0" 

6'  10" 

7'0" 

1'  SO" 

8'0" 

8'  SO" 

9'  0" 

9'  SO" 

10'  0" 

10'  SO" 

11'  O1' 

ll'IO" 

5°     0' 

9.9951 

9.9947 

9.9944 

9.9940 

9.9937 

9.9933 

9.9929 

9.9926 

9.9922 

9.9919 

10 

9.9953 

9.9949 

9.9946  '9.9942 

9.9939 

9.9935  9.9932 

9.9928 

9.9925 

9.9921 

20 
30 

9.9959 

9.9954 
9.9956 

9.9951 
9.9952 

9.9948  9.9944 
9.9949  9.9946 

9.9941 
9.9943 

9.9937  9.9934 
9.9939  9.9936 

9.9931 
9.9933 

9.9927 
9.9929 

9.9924 

40 

9.9960 

9.9957 

9.9954 

9.9951 

9.9948 

9.9944 

9.9941 

9.9938 

9.9935 

9.9932 

50 

9.9965 

9.9962 

9.9958 

9.9955  9.9952 

9.9949 

9.9946 

9.9943 

9.9940 

9.9937 

6       0 

9.9966  9.9963 

9.9960 

9.9957  9.9954 

9.9951 

9.9948 

9.9945 

9.9942 

9.9939 

20 

9.9968  9.9965 

9.9962 

9.9959  9.9956 

9.9954 

9.9951 

9.9948 

9.9945 

40 
7       0 

9.9969  9.9967 
9.9971  9.9968 

9.9964 
9.9966 

9.9961  9.9959 
9.9963  9.9961 

9.9956 
9.9958 

9.9953 
9.9956 

9.9951 
9.9953 

9.9948 

I 

20 

9.9972  9.9970 

9.9968 

9.9965  9.9963 

9.9960 

9.9958 

40 

9.9974  9.9971 

9.9969 

9.9967  9.9965 

9.9962 

8       0 

9.9975  9.9973  9.9971 

9.9968 

9.9966 

9.9964 

20 

9.9976  9.9974 

9.9972 

9.9970 

9.9968 

40 

9.9977 

9.9975 

9.9973 

9.9971 

9       0 

9.9978 

9.9976 

9.9974 

9.9972 

20 

9.9979  9.9977 

9.9975 

40 

9.9980  9.9978 

9.9976 

10 

9.9981 

9.9979 

9.9977 

11 

9.9983 

9.9981 

12 

9.9985 

13 

14 

15 

16 

18 

20 

25 

30 

50 

90 

LOG.    C. 

App.  Alt. 

REDUCED  REFRACTION  AND  PARALLAX  OF  Q  OR  #. 

of 

©or-)f  • 

6'0" 

6'  10" 

7/0// 

7'  SO" 

8'0" 

8'  SO" 

9'0" 

9'  10" 

10'  0" 

10'  SO' 

ll'O' 

11'  SO" 

5°     0' 

9.9949 

9.9946 

9.9942 

9.9938 

9.9935 

9.9931 

9.992-7 

9.9924 

9.9920 

9.9916 

20 

9.9956 

9.9953 

9.9949 

9.9946 

9.9942 

9.9939 

9.9936 

9.9932 

9.9929 

9.9925 

9.9922 

40 

9.9962 

9.9959 

9.9955 

9.9952 

9.9949 

9.9946 

9.9943 

9.9939 

9.9936 

9.9933 

9.9930 

6       0 

9.9964 

9.9961 

9.9958 

9.9955 

9.9952 

9.9949 

9.9946 

9.9943 

9.9940 

9.9937 

20 

9.9966 

9.9963 

9.9960 

9.9957 

9.9955 

9.9952 

9.9949 

9.9946 

9.9943 

40 

9.9968 

9.9965 

9.9962 

9.9960 

9.9957 

9.9954 

9.9951 

9.9949 

9.9946 

7 

9.9969 

9.9967 

9.9964 

9.9962 

9.9959 

9.9956 

9.9954 

9.9951 

8 

9.9973 

9.9971 

9.9969 

9.9966 

9.9964 

9.9962 

9.9960 

9 

9.9976 

9.9974 

9.9972 

9.9970 

9.9968 

10 

9.9979 

9.9977 

9.9975 

11 

9.9981 

9.9979 

12 

9.9983 

13 

14 

15 

16 

17 

18 

20 

25 

30 

40 

50 

90 

31 


TABLE    V.    LOG.  D. 

Logs.  A,  B,  C,  and  D,  for  Lunars. 


A?F 

REDUCED  PARALLAX  AND  REFRACTION  OF  > 

Alt. 
of  2>. 

41' 

42' 

43' 

44' 

45.' 

46' 

47' 

48' 

49' 

5O' 

51' 

52' 

53' 

54' 

55' 

5°0' 

0283 

0290 

0296 

0303 

0310 

0316 

0323 

0329 

0336 

0343 

0349 

0356 

0362 

0369 

3 

0280 

0287 

0293 

0300 

0307 

0313 

0320 

0326 

0333 

0339 

0346 

0352 

0359 

0365 

6 

0277 

0284 

0291 

0297 

0304 

0310 

0317 

0323 

0330 

0336 

0342 

0349 

0355 

0362 

9 

0275 

0281 

0288 

0294 

0301 

0307 

0313 

0320 

0326 

0333 

0339 

0345 

0352 

0358 

12 

0272 

0279 

0285 

0291 

0298 

0304 

0310 

0317 

0323 

0330 

0336 

0342 

0349 

0355 

5  15 

0270 

0276 

0282 

0289 

0295 

0301 

0308 

0314 

0320 

0326 

0333 

0339 

0345 

0351 

18 

0267 

0273 

0280 

0286 

0292 

0298 

0305 

0311 

0317 

0323 

0330 

0336 

0342 

0348 

21 

0264 

0271 

0277 

0283 

0289 

0296 

0302 

0308 

0314 

0320 

0327 

0333 

0339 

0345 

24 

0262 

0268 

0274 

0281 

0287 

0293 

0299 

0305 

0311 

0317 

0324 

0330 

0336 

0342 

27 

0260 

0266 

0272 

0278 

0284 

0290 

0296 

0302 

0308 

0314 

0321 

0327 

0333 

0339 

530 

0257 

0263 

0269 

0275 

0282 

0288 

0294 

0300 

0306 

0312 

0318 

0324 

0330 

0336 

33 

0255 

0261 

0267 

0273 

0279 

0285 

0291 

0297 

0303 

0309 

0315 

0321 

0327 

0333 

36 

0253 

0259 

0265 

0271 

0276 

0282 

0288 

0294 

0300 

0306 

0312 

0318 

0324 

0330 

39 

0256 

0262 

0268 

0274 

0280 

0286 

0292 

0298 

0303 

0309 

0315 

0321 

0327 

42 

0254 

0260 

0266 

0272 

0277 

0283 

0289 

0295 

0301 

0306 

0312 

0318 

03*24 

545 

0252 

0258 

0263 

0269 

0275 

0281 

0287 

0292 

0298 

0304 

0310 

0315 

0321 

48 

0250 

0255 

0261 

0267 

0273 

0278 

0284 

0290 

0295 

0301 

0307 

0313 

0318 

51 

0247 

0253 

0259 

0265 

0270 

0276 

0282 

0287 

0293 

0299 

0304 

0310 

0316 

54 

0245 

0251 

0257 

0262 

0268 

0274 

0279 

0285 

0290 

0296 

0302 

0307 

0313 

57 

0243 

0249 

0254 

0260 

0266 

0271 

0277 

0282 

0288 

0294 

0299 

0305 

0310 

6  0 

0241 

0247 

0252 

0258 

0263 

0269 

0275 

0280 

0286 

0291 

0297 

0302 

0308 

3 

0239 

0245 

0250 

0256 

0261 

0267 

0272 

0278 

0283 

0289 

0294 

0300 

0305 

6 

0237 

0243 

0248 

0254 

0259 

0265 

0270 

0275 

0281 

0286 

0292 

0297 

0302 

9 

0235 

0241 

0246 

0252 

0257 

0262 

0268 

0273 

0279 

0284 

0289 

0295 

0300 

12 

0233 

0239 

0244 

0249 

0255 

0260 

0266 

0271 

0276 

0282 

0287 

0292 

0298 

6  15 

0231 

0237 

0242 

0247 

0253 

0258 

0263 

0269 

0274 

0279 

0285 

0290 

0295 

18 

0230 

0235 

0240 

0245 

0251 

0256 

0261 

0267 

0272 

0277 

0282 

0288 

0293 

21 

0228 

0233 

0238 

0243 

0249 

0254 

0259 

0264 

0270 

0275 

0280 

0285 

0290 

24 

0226 

0231 

0236 

0242 

0247 

0252 

0257 

0262 

0267 

0273 

0278 

0283 

0288 

27 

0229 

0234 

0240 

0245 

0250 

0255 

0260 

0265 

0271 

0276 

0281 

0286 

0291 

630 

0227 

0233 

0238 

0243 

0248 

0253 

0258 

0263 

0268 

0274 

0279 

0284 

0289 

33 

0226 

0231 

0236 

0241 

0246 

0251 

0256 

0261 

0266 

0271 

0276 

0281 

0287 

36 

0224 

0229 

0234 

0239 

0244 

0249 

0254 

0259 

0264 

0269 

0274 

0279 

0284 

39 

0222 

0227 

0232 

0237 

0242 

0247 

0252 

0257 

0262 

0267 

0272 

0277 

0282 

42 

0220 

0225 

0230 

0235 

0240 

0245 

0250 

0255 

0260 

0265 

0270 

0275 

0280 

645 

0219 

0224 

0229 

0234 

0239 

0244 

0248 

0253 

0258 

0263 

0268 

0273 

0278 

48 

0217 

0222 

0227 

0232 

0237 

0242 

0247 

0251 

0256!  0261 

0266 

0271 

0276 

51 

0216 

0220 

0225 

0230 

0235 

0240 

0245 

0250 

0254  0259 

0264 

0269 

0274 

54 

0214 

0219 

0224 

0228 

0233 

0238 

0243 

0248;  0253  0257 

0262 

0267 

1272 

57 

0212 

0217 

0222 

0227 

0232 

0236 

0241 

0246  0251  0255 

0260 

0265 

0270 

7  0 

0211 

0216 

0220 

0225 

0230 

0235 

0239 

0244  !  0249,  0254 

0258 

0263 

0268 

3 

0209 

0214 

0219 

0223 

0228 

0233 

0238 

0242  0247  0252 

0256 

0261 

0266 

6 

0208 

0212 

0217 

0222 

0227 

0231 

0236 

0241  0245  0250 

0255 

0259 

0264 

9 

0211 

0216 

0220 

0225 

0230 

0234 

0239  0243  !  0248 

0253 

0257 

0262 

12 

0209 

0214 

0219 

0223 

0228 

0232 

0237  j  0242  0246 

0251 

0255 

0260 

7  15 

0208 

0212 

0217 

0222 

0226 

0231 

0235  0240  0245 

0249 

0254 

0258 

18 

0206 

0211 

0216 

0220 

0225 

0229 

0234  0238  0243 

0247 

0252 

0256 

21 

0205 

0209 

0214 

0219 

0223 

0228 

0232  0237;  0241 

0246 

0250 

0255 

24 

0204 

0208 

0213 

0217 

0222 

0226 

0230  0235  0239 

0244 

0248 

0253 

27 

0202 

0207 

0211 

0216 

0220 

0224 

0229;  0233  0238 

0242 

0247 

0251 

730 

0201 

0205 

0210 

0214 

0218 

0223 

0227  0232!  0236 

0241 

0245 

0249 

33 

0199 

0204 

0208 

0213 

0217 

0221 

0226 

0230  0234 

0239 

0243 

0248 

36 

0198 

0202 

0207 

0211 

0215 

0220 

0224 

0229  0233 

0237 

0242 

0246 

39 

0197 

0201 

0205 

0210 

0214 

0218 

0223  0227  0231 

0236 

0240 

0244 

42 

0195 

0200 

0204 

0208 

0213 

0217 

0221  0225  0230 

0234 

0238 

0243 

7  45 

0194 

0198 

0203 

0207 

0211 

0215 

0220!  0224  0228 

0232 

0237 

0241 

48 

0193 

0197 

0201 

0205 

0210 

0214 

0218  0222;  0227 

0231 

0235 

0239 

51 

019 

0196 

0200 

0204 

0208 

0213 

0217 

0221  0225 

0229 

0234 

0238 

54 

0190 

0194 

0198 

0203 

0207 

0211 

0215 

0219  0224 

0228 

0232 

0236 

57 

0189 

0193 

0197 

0201 

0206 

0210 

0214 

0218  0222 

0226 

0230 

0235 

*  0 

0188 

0192 

0196 

020C 

0204 

0208 

0212 

0217  0221 

0225 

0229 

0233 

32 


TABLE    V.    LOG.  D. 

Logs.  A,  B,  C,  and  D,  for  Lunars. 


App. 

A  1* 

EEDUCED  PARALLAX  AND  REFRACTION  OF  }>. 

Alt. 
of  > 

45' 

46' 

47' 

48' 

49' 

50' 

51' 

52' 

53' 

54' 

55' 

56' 

57' 

58 

8D0' 

.0192 

0196 

0200 

0204 

0208 

0212 

0217 

0221 

0225 

0229 

0233 

0237 

5 

.0190 

0194 

0198 

0202 

0206 

0210 

0214 

0218 

0222 

0227 

0231 

0235 

10 

.0188 

0192 

0196 

0200 

0204 

0208 

0212 

0216 

0220 

0224 

0228 

0232 

15 

.0186 

0190 

0194 

0198 

0202 

0206 

0210 

0214 

0218 

0222 

0226 

0230 

20 

.0184 

0188 

0192 

0196 

0200 

0204 

0207 

0211 

0215 

0219 

0223 

0227 

25 

.0182 

0186 

0190 

0194 

0197 

0201 

0205 

0209 

0213 

0217 

0221 

0225 

8  30 

.0180 

0184 

0188 

0192 

0195 

0199 

0203 

0207 

0211 

0215 

0219 

0223 

35 

.0178 

0182 

0186 

0190 

0193 

0197 

0201 

0205 

0209 

0213 

0216 

0220 

40 

.0176 

0180 

0184 

0188 

0191 

0195 

0199 

0203 

0207 

0210 

0214 

0218 

45 

.0174 

0178 

0182 

0186 

0189 

0193 

0197 

0201 

0205 

0208 

0212 

0216 

50 

.0173 

0176 

0180 

0184 

0188 

0191 

0195 

0199 

0202 

0206 

0210 

0214 

55 

.0171 

0175 

0178 

0182 

0186 

0189 

0193 

0197 

0200 

0204 

0208 

0212 

9  0 

.0169 

0173 

0177 

0180 

0184 

0188 

0191 

0195 

0198 

0202 

0206 

0209 

5 

.0167 

0171 

0175 

0178 

0182 

0186 

0189 

0193 

0197 

0200 

0204 

0207 

10 

.0166 

0169 

0173 

0177 

0180 

0184 

0187 

0191 

0195 

0198 

0202 

0205 

15 

.0164 

0168 

0171 

0175 

0179 

0182 

0186 

0189 

0193 

0196 

0200 

0203 

20 

.0163 

016610170 

0173 

0177 

0180 

0184 

0187 

0191 

0194 

0198 

0201 

25 

.0161 

0165 

0168 

0172 

0175 

0179 

0182 

0186 

0189 

0193 

0196 

0199 

9  30 

0163 

0166 

0170 

0173 

0177 

0180 

0184 

0187 

0191 

0194 

0198 

35 

0161 

0165 

0168 

0172 

0175 

0179 

0182 

0185 

0189 

0192 

0196 

- 

40 

0160 

0163 

0167 

0170 

0174 

0177 

0180 

0184 

0187 

0191 

0194 

45 

0158 

0162 

0165 

0169 

0172 

0175 

0179 

0182 

0185 

0189 

0192 

0195 

50 

0157J  0160 

0164 

0167 

0170 

0174 

0177 

0180 

0184 

0187 

0190 

0194 

55 

0156  0159 

0162 

0165 

0169 

0172 

0175 

0179 

0182 

0185 

0189 

0192 

10  0 

0154 

0157 

0161 

0164 

0167 

0171 

0174 

0177 

0180 

0184 

0187 

0190 

5 

0153 

0156 

0159 

0162 

0166 

0169 

0172 

0175 

0179 

0182 

0185 

0188 

10 

0151 

0155 

0158 

0161. 

0164 

0167 

0171 

0174 

0177 

0180 

0183 

0187 

15 

0150 

0153 

0156 

0160 

0163 

0166 

0169 

0172 

0175 

0179 

0182 

0185 

20 

0149 

0152 

0155 

0158 

0161 

0164 

0168 

0171 

0174 

0177 

0180 

0183 

25 

0147 

0150 

0154 

0157 

0160 

0163 

0166 

0169 

0172 

0175 

0179 

0182 

1030 

0146 

0149 

0152 

0155 

0158 

0162 

0165 

0168 

0171 

0174 

0177 

0180 

35 

0145 

0148 

0151 

0154 

0157 

0160 

0163 

0166 

0169 

0172 

0175 

0179 

40 

. 

0143 

0147 

0150 

0153 

0156 

0159 

0162 

0165 

0168  0171  0174 

0177 

45 

0142 

0145 

0148 

0151 

0154 

0157 

0160 

0163 

0166  0169  0172 

0175 

50 

0141 

0144 

0147 

0150 

0153 

0156 

0159 

0162 

0165  0168  0171 

0174 

55 

0140 

0143 

0146 

0149 

0152 

0155 

0158 

0161 

0164  0167  0170 

0172 

11  0 

0139 

0142 

0145 

0147 

0150 

0153 

0156 

0159 

0162  0165-1)168 

0171 

5 

0137 

0140 

0143 

0146 

0149 

0152 

0155 

0158 

0161  0164  0167 

0170 

10 

0139 

0142 

0145 

0148 

0151 

0154 

0157 

0159  0162  0165 

0168 

15 

0138 

0141 

0144 

0147 

0150 

0152 

0155 

0158  0161  0164 

0167 

20 

0137 

0140 

0143 

0145 

0148 

0151 

0154 

0157  0160  0163 

0165 

25 

0136 

0139 

0141 

0144 

0147 

0150 

0153 

0156  0158  0161 

0164 

1130 

0135 

0137 

0140 

0143 

0146 

0149 

0151 

0154  0157  0160 

0163 

35 

0133 

0136 

0139 

0142 

0145 

0147 

0150 

0153  0156  0159 

0161 

40 

0132 

0135 

0138 

0141 

0143 

0146 

0149 

0152i  0154  0157 

0160 

45 

0131 

0134 

0137 

0140 

0142 

0145 

0148 

0150  0153  0156 

0159 

50 

0130 

0133 

0136 

0138 

0141 

0144 

0147 

0149  0152  0155 

0157 

55 

0129 

0132 

0135 

0137 

0140 

0143 

0145 

0148 

0151  0153 

0156 

12  0 

0128 

0131 

0134 

0136 

0139 

0142 

0144 

0147 

0150  0152 

0155 

5 

0127 

0130 

0132 

0135 

0138 

0140 

0143 

0146 

0148  '  0151 

0154 

10 

0126 

0129 

0131 

0134 

0137 

0139 

0142 

0145J  0147  0150 

0152 

15 

0125 

0128 

0130 

0133 

0136 

0138 

0141 

0143  0146  0149 

0151 

20 

0124 

0127 

0129 

0132 

0135 

0137 

0140 

0142  0145  0147 

0150 

25 

0123 

0126 

0128 

0131 

0133 

0136 

0139 

0141  0144J  0146 

0149 

1230 

0122 

0125 

0127 

0130 

0132 

0135 

0138 

0140 

0143  0145 

0148 

. 

35 

0121 

0124 

0126 

0129 

0131 

0134 

0136 

0139 

0141  0144 

0147 

40 

0120 

0123 

0125 

0128 

0130 

0133 

0135 

0138 

0140 

0143 

0145 

45 

0119 

0122 

0124 

0127 

0129 

0132 

0134 

0137 

0139 

0142 

0144 

0147 

50 

0118 

0121 

0123 

0126 

0128 

0131 

0133 

0136 

0138 

0141 

0143 

0146 

55 

0118 

0120 

0123 

0125 

0127 

0130 

0132 

0135 

0137 

0140 

0142 

0145 

13  0 

0117 

0119 

0122 

0124 

0126 

0129 

0131 

0134 

0136 

0139 

0141 

0143 

33 


TABLE    V.    LOG.  D. 

Logs.  A,  B,  C,  and  D,  for  Lunars. 


l?tp' 

REDUCED  PARALLAX  AND  REFRACTION  OF  D. 

of  3>- 

4T 

48' 

49' 

.*O 

51' 

52' 

53' 

54' 

55' 

56' 

57' 

58' 

59' 

13°  0' 

.0117 

0119 

0122 

0124 

0126 

0129 

0131 

0134 

0136 

0139 

0141 

0143 

10 

.0115 

0117 

0120 

0122 

0125 

0127 

0129 

0132 

0134 

0137 

0139 

0141 

20 

.0113 

0116 

0118 

0120 

0123 

0125 

0127 

0130 

0132 

0134 

0137 

0139 

30 

.0112 

0114 

0116 

0119 

0121 

0123 

0125 

0128 

0130 

0132 

0135 

0137 

40 

0112 

0114 

0117 

O119 

0121 

0124 

0126 

0128 

0131 

0133 

0135 

50 

0111 

0113 

0115 

0117 

0120 

0122 

0124 

0126 

0129 

0131 

0133 

14  0 

0109 

0111 

0113 

0116 

0118 

0120 

0122 

0125 

0127 

0129 

0131 

10 

0107 

0110 

0112 

0114 

0116 

0118 

0121 

0123 

0125 

0127 

0129 

20 

0106 

0108 

0110 

0112 

0114 

0117 

0119 

0121 

0123 

0125 

0127 

30 

0104 

0106 

0109 

0111 

0113 

0115 

0117 

0119 

0121 

0123 

0126 

40 

0103 

0105 

0107 

0109 

0111 

0113 

0115 

0118 

0120 

0122 

0124 

50 

0101 

0103 

0106 

0108 

0110 

0112 

0114 

0116 

0118 

0120 

0122 

15  0 

0100 

0102 

0104 

0106 

0108 

0110 

0112 

0114 

0116 

0118 

0120 

10 

0099 

0101 

0103 

0105 

0107 

0109 

0111 

0113 

0115 

0117 

0119 

20 

0097 

0099 

0101 

0103 

0105 

0107 

0109 

0111 

0113 

0115 

0117 

30 

0096 

0098 

0100 

0102 

0104 

0106 

0108 

0110 

0112 

0113 

0115 

40 

0094 

0096 

0098 

0100 

0102 

0104 

0106 

0108 

0110 

0112 

0114 

50 

0093 

0095 

0097 

0099 

0101 

0103 

0105 

0107 

0108 

0110 

0112 

16  0 

0092 

0094 

0096 

0098 

0099 

0101 

0103 

0105 

0107 

0109 

0111 

10 

0091 

0093 

0094 

0096 

0098 

0100 

0102 

0104 

0106 

0107 

0109 

20 

0089 

0091 

0093 

0095 

0097 

0099 

0100 

0102 

0104 

0106 

0108 

30 

0088 

0090 

0092 

0094 

0096 

0097 

0099 

0101 

0103 

0105 

0106 

40 

0087 

0089 

0091 

0092 

0094 

0096 

0098 

0100 

0101 

0103 

0105 

50 

0086 

0088 

0089 

0091 

0093 

0095 

0096 

0098 

0100 

0102 

0104 

17  0 

0085 

0087 

0088 

0090 

0092 

0093 

0095 

0097 

0099 

0100 

0102 

10 

0084 

0085 

0087 

0089 

0091 

0092 

0094 

0096 

0097 

0099 

0101 

20 

0083 

0084 

0086 

0088 

0089 

0091 

0093 

0094 

0096 

0098 

0099 

30 

0083 

0085 

0086 

0088 

0090 

0091 

0093 

0095 

0096 

0098 

40 

0082 

0084 

0085 

0087 

0089 

0090 

0092 

0094 

0095 

0097 

50 

0081 

0083 

0084 

0086 

0087 

0089 

0091 

0092 

0094 

0096 

18  0 

0080 

0082 

0083 

0085 

0086 

0088 

0090 

0091 

0093 

0094 

20 

0078 

0079 

0081 

0083 

0084 

0086 

0087 

0089 

0090 

0092 

0093 

40 

0076 

0077 

0079 

0080 

0082 

0083 

0085 

0087 

0088 

0090 

0091 

19  0 

0074 

0075 

0077 

0078 

0080 

0081 

0083 

0084 

0086 

0087 

0089 

20 

0072 

0073 

0075 

0076 

0078 

0079 

0081 

0082 

0084 

0085 

0086 

40 

0070 

0072 

0073 

0074 

0076 

0077 

0079 

0080 

0081 

0083 

0084 

20  0 

0068 

0070 

0071 

0073 

0074 

0075 

0077 

0078 

0079 

0081 

0082 

20 

0067 

0068 

0069 

0071 

0072 

0073 

0075 

0076 

0077 

0079 

0080 

40 

0065 

0066 

0068 

0069 

0070 

0072 

0073 

0074 

0075 

0077 

0078 

21  0 

0063 

0065 

0066 

0067 

0068 

0070 

0071 

0072 

0074 

0075 

0076 

20 

0062 

0063 

0064 

0065 

0067 

0068 

0069 

0070 

0072 

0073 

0074 

40 

0060 

0061 

0063 

0064 

0065 

0066 

0067 

0069 

0070 

0071 

0072 

22  0 

0059 

0060 

0061 

0062 

0063 

0065 

0066 

0067 

0068 

0069 

0070 

20 

0057 

0058 

0059 

0061 

0062 

0063 

0064 

0065 

0066 

0068 

0069 

40 

0056 

0057 

0058 

0059 

0060 

0061 

0062 

0064 

0065 

0066 

0067 

23  0 

0054 

0055 

0057 

0058 

0059 

0060 

0061 

0062 

0063 

0064 

0065 

20 

0053 

0054 

0055 

0056 

0057 

0058 

0059 

0060 

0061 

0063 

0064 

40 

0052 

0053 

0054 

0055 

0056 

0057 

0058 

0059 

0060 

0061 

0062 

24  0 

0050 

0051 

0052 

0053 

0054 

0055 

0056 

0057 

0058 

0059 

0060 

20 

0050 

0051 

0052 

0053 

0054 

0055 

0056 

0057 

0058 

0059 

40 

0049 

0050 

0051 

0052 

0053 

0053 

0054 

0055 

0056 

0057 

25  0 

0047 

0048 

0049 

0050 

0051 

0052 

0053 

0054 

0055 

0056 

20 

0046 

0047 

0048 

0049 

0050 

0051 

0052 

0053 

0053 

0054 

40 

0045 

0046 

0047 

0048 

0049 

0049 

0050 

0051 

0052 

0053 

26  0 

0044 

0045 

0046 

0046 

0047 

0048 

0049 

0050 

0051 

0052 

20 

0043 

0043 

0044 

0045 

0046 

0047 

0048 

0048 

0049 

0050 

40 

0041 

0042 

0043 

0044 

0045 

0046 

0046 

0047 

0048 

0049 

27  0 

0040 

0041 

0042 

0043 

0044 

0044 

0045 

0046 

0047 

0047 

20 

0039 

0040 

0041 

0042 

0042 

0043 

0044 

0045 

0045 

0046 

40 

0038 

0039 

0040 

0040 

0041 

0042 

0043 

0043 

0044 

0045 

28  0 

0037 

0038 

0039 

0039 

0040 

0041 

0042 

0042 

0043 

0044 

34 


TABLE    V.    LOG.  D. 

Logs.  A,  B,  C,  and  D,  for  Limars. 


Apparent 
Altitude 
of  5- 

REDUCED  PARALLAX  AND  REFRACTION  OF  > 

50' 

51' 

52' 

53' 

54' 

55' 

56' 

57' 

58' 

59' 

6O' 

28°  0' 

0.0037 

0.0038 

0.0039 

0.0039 

0.0040 

0.0041 

0.0042 

0.0042 

0.0043 

0.0044 

30 

0.0036 

0  0036  !  0.0037 

0.0038 

0.0038 

0.0039 

0.0040 

0.0040:0.0041 

0.0042 

29   0 

0.0034 

0.0035  0.0035 

0.0036 

0.0037 

0.0037 

0.0038 

0.0039  0.0039 

0.0040 

30 

0.0033  0.0033  0.0034 

0.0035 

0.0035 

0.0036 

0.0036 

0.0037  0.0038 

0.0038 

30  0 

0.003  1 

0.0032  0.0032 

0.0033 

0.0034 

0.0034 

0.0035 

0.0035  0.0036 

0.0037 

30 

0.0030 

0.0030  00031 

0.0031 

0.0032 

0.0033 

0.0033 

0.0034  0.0034 

0.0035 

31   0 

0.0028 

0.0029  0.0029 

00030 

0.0031 

0.0031 

0.0032 

0.0032  0.0033 

0.0033 

30 

0.0027 

0.0028  0.0028 

0.0029 

0.0029 

0.0030 

0.0030 

0.0031  0.0031 

0.0032 

0.0032 

32   0 

0.0026 

0.0026  0.0027 

0.0027 

0.0028 

0.0028 

0.0029 

0.0029  0.0030 

0.0030 

0.0031 

30 

00024 

0.0025  0.0025 

0.0026 

0.0026 

0.0027 

0.0027 

0.0028  0.0028 

0.0029 

0.0029 

33   0 

0.0023 

0.0024  0.0024 

0.0025 

0.0025 

0.0025 

0.0026 

0.0026  0.0027 

0.0027 

0.0028 

30 

0.0022 

0.0022  0.0023 

0.0023 

0.0024 

0.0024 

0.0025 

0.0025 

00025 

0.0026 

0.0026 

34   0 

0.0021 

0.0021  0.0022 

0.0022 

0.0022 

0.0023 

0.0023 

0.0024 

0.0024 

0.0024 

0.0025 

30 

0.0020 

0.0020  0.0020 

0.0021 

0.0021 

0.0022 

0.0022 

0.0022 

0.0023 

0.0023 

0.0023 

35   0 

0.0018 

00019.0.0019 

0.0020 

0.0020 

0.0020 

0.0021 

0.0021 

0.0021 

0.0022 

0.0022 

30 

0.0017 

0.0018  0.0018 

0.0018 

0.0019 

0.0019 

0.0019 

0.0020 

0.0020 

0.0020 

0.0021 

36   0 

0.0016 

0.0017  0.0017 

0.0017 

0.0018 

0.0018 

0.0018 

0.0019 

0.0019 

0.0019 

0.0019 

30 

0.0015 

0.0016  0.0016 

0.0016 

0.0016 

0.0017 

0.0017 

0.0017 

0.0018 

0.0018 

0.0018 

37   0 

0.0014 

0.0014  0.0015 

0.0015 

0.0015 

0.0016 

0.0016 

0.0016 

0.0016 

0.0017 

0.0017 

30 

0.0013 

0.0013  0.0014 

0.0014 

0.0014 

0.0014 

0.0015 

00015 

0.0015 

0.0015 

0.0016 

38  0 

0.0012 

0.0012  0.0013 

0.0013 

0.0013 

0.0013 

0.0014 

0.0014 

0.0014 

0.0014 

0.0014 

30 

0.0011 

0.0011  0.0012 

0.0012 

0.0012 

0.0012 

0.0012 

0.0013 

0.0013 

0.0013 

0.0013 

39  0 

0.0010 

0.0010  0.0011 

0.0011 

0.0011 

0.0011 

00011 

0.0012 

0.0012 

0.0012 

0.0012 

30 

0.0009  0.0010 

0.0010 

0.0010 

0.0010 

0.0010 

0.0010 

0.0011 

0.0011 

0.0011 

40 

0.0008  0.0009 

0.0009 

0.0009 

0.0009 

00009 

0.0009 

0.0010 

0.0010 

0.0010 

41 

0.0007  0.0007 

0.0007 

0.0007 

0.0007 

0.0007 

0.0007 

00007 

0.0008 

0.0008 

42 

0.0005  0.0005 

0.0005 

0.0005 

0.0005 

0.0005 

0.0005 

0.0005 

0.0005 

0.0006 

43 

0.0003  0.0003 

0.0003 

0.0003 

0.0003 

0.0003 

0.0003 

0.0003 

0.0003 

0.0004 

44 

0.0001  0.0001 

0.0001 

0.0001 

0.0001 

0.0001 

0.0002 

0.0002 

0.0002 

0.0002 

45 

00000  0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

46 

9.9998  9.9998 

9.9998 

9.9998 

9.9998 

9.9998 

9.9998 

9.9998 

99998 

9.9998 

47 

9.9997  9.9997 

9.9997 

9.9997 

9.9996 

9.9996 

9.9996 

9.9996 

9.9996 

9.9996 

48 

9.999519.9995 

9.9995 

9.9995 

9.9995 

9.9995 

9.9995 

9.9995 

9.9994 

9.9994 

49 

9.999419.9994 

9.9994 

9.9993 

9.9993 

9.9993 

9.9993 

9.9993 

9.9993 

9.9993 

50 

9.9992 

9.9992 

9.9992 

9.9992 

9.9992 

9.9992 

9.9992 

9.9991 

9.9991 

9.9991 

51 

9.9991 

9.9991 

9.9991 

9.9991 

9.9990 

9.9990 

9.9990 

9.9990 

9.9990 

9.9990 

52 

9.9990 

9.9990 

9.9990 

9.9989 

9.9989 

9.9989 

9.9989 

9.9989 

9.9988 

9.9988 

53 

9.9989 

9.9988 

9.9988 

9.9988 

9.9988 

9.9988 

9.9987 

9.9987 

9.9987 

9.9987 

54 

9.9988 

9.9987 

9.9987 

9.9987 

9.9987 

9.9986 

9.9986 

9.9986 

9.9986 

9.9985 

55 

9.9986 

9.9986 

9.9986 

9.9986 

9.9985 

9.9985 

9.9985 

9.9984 

9.9984 

9.9984 

56 

9.9985 

9.9985 

9.9985 

9.9984 

9.9984 

9.9984 

9^9984 

9.9983 

9.9983 

9.9983 

57 

9.9984 

9.9984 

9.9984 

9.9983 

9.9983 

9.9983 

9.9982 

9.9982 

9.9982 

9.9981 

58 

9.9983 

9.9983 

9.9983 

9.9982 

9.9982 

9.9982 

9.9981 

9.9981 

9.9981 

9.9980 

59 

9.9982 

9.9982 

9.9981 

9.9981 

9.9981 

9.9980 

9.9980 

9.9980 

9.9979 

9.9979 

60 

9.9981 

9.9981 

9.9980 

9.9980 

9.9980 

9.9979 

9.9979 

9.9979 

9.9978 

9.9978 

61 

9.9980 

9.9980 

9.9980 

9.9979 

9.9979 

9.9978 

9.9978 

9.9978 

9.9977 

9.9977 

62 

9.9979 

9.9979 

9.9979 

9.9978 

9.9978 

9.9977 

9.9977 

9.9977 

9.9976 

9.9976 

63 

9.9979 

9.9978 

9.9978 

9.9977 

9.9977 

9.9.976 

9.9976 

9.9976 

9.9975 

9.9975 

64 

9.9978 

9.9977 

9.9977 

9.9976 

9.9976 

9.9976 

9.9975 

9.9975 

9.9974 

9.9974 

65 

9.9977 

9.9977 

9.9976 

9.9976 

9.9975 

9.9975 

9.9974 

9.9974 

9.9973 

9.9972 

66 

9.9976 

9.9976 

9.9975 

9.9975 

9.9974 

9.9974 

9.9973 

9.9973 

9.9973 

9.9972 

67 

9.9976 

9.9975 

9.9975 

9.9974 

9.9974 

9.9973 

9.9973 

9.9972 

9.9972 

9.9971 

68 

9.9975 

9.9974 

9.9974 

9.9973 

9.9973 

9.9972 

9.9972 

9.9971 

9.9971 

9.9970 

69 

9.9974 

9.9974 

9.9973 

9.9973 

9.9972 

9.9972 

9.9971 

9.9971 

9.9970  9.9970 

70 

9.9974 

9.9973 

9.9973 

9.9972 

9.9972 

9.9971 

9.9970 

9.9970 

9.9969  9.9969 

72 

9.9972 

9.9972 

9.9971 

9.9971 

9.9970 

9.9970 

9.9969 

9.9969 

9.9968 

9.9968 

74 

9.9971 

9.9971 

9.9970 

9.9970 

9.9969 

9.9969 

9.9968 

9.9968 

9.9967 

9.9966 

76 

9.9971 

9.9970 

9.9969 

9.9969 

9.9968 

9.9968 

9.9967 

9.9966 

9.9966(9.9965 

78 

9.9970 

9.9969 

9.9969 

9.9968 

9.9967 

9.9967 

9.9966 

9.9966 

9.996519.9964 

80 

9.9969 

9.9969 

9.9968 

9.9967 

9.9967 

9.9966 

9.9965 

9.9965 

9.9964 

9.9964 

90 

9.9968  9.9967 

9.9966 

9.9966 

9.9965 

9.9964 

9.9964 

9.9963 

9.9963  9.9962 

35 


TABLE    VI. 

'Second  Correction  of  the  Lunar  Distance. 


Appar- 

FIRST   CORRECTION    OF    DISTANCE. 

Appar- 

ent 

ent 

Dis- 

Dis- 

tance. 

8' 

/ 

1(1  1J 

IHG 

18 

J4 

M 

27;  28 

29  307 

31 

3J 

13 

ii 

Itf 

X 

tance. 

^0 

2122 

2i 

24 

Subtr. 

Add. 

;i5°o' 

b 

2 

3 

5 

C) 

8 

11 

13 

14  16 

17 

19 

20  22 

24 

'26 

27  29 

33 

33 

35 

38    40 

42 

30 

0 

2 

3 

5 

6 

8 

10 

13 

14 

15  17 

18 

20  21 

23 

25 

26  28  32 

32 

34 

36    39  i   41 

16    0 

0 

3 

4 

6|  8 

10 

12 

13 

15  16 

18 

19  21 

22 

24 

26  27 

31 

31 

33 

35    37 

39 

30 

0 

3 

4 

6    8 

10 

12 

13 

14  16 

17 

18  20 

21 

23 

25  27 

30 

30 

32 

34!   36 

38 

17    0 

0 

3 

4 

6!   7 

9 

11 

13 

14 

15 

16 

18 

19 

21 

22 

24  26 

29 

29 

31 

33    35 

37 

30 

0 

3 

4 

5 

7 

9 

11 

12  13 

15 

16 

17 

19 

20 

22 

23  25 

28 

28 

30 

32    34    36 

18    0 

0 

3 

4 

5 

7 

9 

11 

12J13 

14 

15 

17 

18 

20 

21 

23  24 

28 

28 

29 

31     33     35 

30 

0 

3 

4 

5 

7 

8 

10 

12 

13 

14 

15 

16 

18 

19 

20 

22i23 

27 

27 

28 

30i   32 

34 

19    0 

0 

3 

4 

5 

6 

8 

10 

11 

12 

13 

15 

16 

17 

13 

20 

21  23 

26 

26 

28 

29    31 

33 

30 

0 

2 

4 

5 

6 

8 

10 

11 

12 

13 

14 

15 

17 

18 

19 

21 

22 

25 

25 

27 

28    30 

32 

20    0 

0 

2 

3 

5 

6 

8 

10 

11 

12 

13 

14 

15 

16 

17 

19 

20;22 

25 

25 

26 

28 

29 

31 

21 

0 

2 

3 

4 

6 

7 

9 

10 

11 

12 

13 

14 

15 

17 

18 

19  20 

23 

23 

25 

26     28  1   29 

22 

0 

2 

3 

4 

6 

7 

9 

10 

10 

11 

12 

14 

15 

16 

17 

18  19 

22 

22 

24 

25     26 

28 

23 

0 

2 

3 

4 

5 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17|19 

21 

21 

22 

24 

25 

27 

24 

0 

2 

3 

4 

5 

6 

8 

9 

9 

10 

11 

12 

13 

14 

15 

ie;i8 

20 

20 

21 

23 

24 

25 

25 

0 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

19 

19 

20 

22    23 

24 

26 

0 

2 

3 

4 

5 

6 

7 

8 

9 

9 

10 

11 

1-2 

13 

14 

15 

1G 

18 

18 

19     21     22 

23 

27 

0 

2 

2 

3 

4 

6 

7 

8 

8 

9 

10 

11 

12 

12 

13 

14 

15 

18 

18 

19 

20  i   21     22 

28 

0 

2 

2 

3 

4 

5 

7 

7 

8 

9 

9 

10 

11 

12 

13 

14 

15  17 

17 

18 

19    20!   21 

29 

0 

2 

2 

3 

4 

5 

6 

7 

8 

8 

9 

id 

11 

11 

12 

13 

14 

16 

16 

17 

18     19  i   20 

30 

0 

2 

2 

3 

4 

5 

6 

7 

7 

8 

9 

9 

10 

11 

12 

13 

14 

15 

15 

16 

171    19     20 

31 

0 

2 

3 

4 

5 

6 

6 

7 

8 

8 

9 

10 

11 

11 

12 

13 

15 

15 

16 

17 

18 

19 

32 

0 

2 

3 

4 

5 

6 

6 

7 

7 

8 

9 

9 

10 

11 

12 

13 

14 

14 

15 

16 

17 

18 

33 

0 

2 

3 

3 

4 

5 

6 

7 

7 

8 

8 

9 

10 

11 

11 

12 

14 

14 

15 

16 

16 

17 

34 

0 

2 

3 

3 

4 

5 

6 

6 

7 

7 

8 

9 

9 

10 

11 

12 

13 

13 

14 

15 

16 

17 

35 

0 

2 

2 

3 

4 

5 

5 

6 

7 

7 

S 

8 

9 

10 

10 

11 

13 

13 

14 

14 

15 

16 

136 

0 

2 

2 

a 

4 

5 

5 

6 

6 

7 

8 

8 

9    9 

10 

11 

12 

12 

13 

14 

15 

16 

37 

0 

2 

2 

3 

4 

B 

'i* 

6 

6 

7 

3 

8 

0 

9 

10 

10 

12 

12 

13 

13 

14 

15 

38 

0 

2 

2 

3 

4 

4 

i?.5 

5 

6 

6 

•:-? 

8 

8 

9 

9 

10 

11 

11 

12 

13 

14 

14 

39 

0 

1 

2 

2 

a 

3 

4 

5 

5 

6 

6 

V? 

7 

8 

0 

9 

10 

11 

11 

12 

12 

13 

14 

40 

0 

1 

2 

2 

3 

3 

4 

5 

5 

6 

6 

7 

7 

8 

Q 

9 

9 

11 

11 

,11 

12 

13 

13 

140° 

42 

0 

0 

2 

2 

3 

4 

4 

5 

5 

6 

6 

7 

71  8 

8 

9 

10 

10 

11 

11 

12 

13 

138 

44 

0 

0 

2 

2 

3 

4 

4 

4 

5 

5 

6 

6 

1 

7 

8 

8 

9 

9 

10J    10 

11 

12 

136 

46 

0 

0 

2 

2 

3 

3 

4 

4 

4 

5 

5 

6 

6 

7 

7 

8 

9 

9 

9     10 

10 

11 

134 

48 

0 

0 

2 

2 

3 

a 

3 

4 

4 

5 

5 

5 

r> 

6 

7 

7 

8 

8 

9 

9 

10 

10 

132 

50 

0 

0 

2 

2 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

7 

8 

8 

8 

8 

9 

9 

130 

52 

0 

0 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

7 

7 

7 

8 

8 

9 

128 

54 

0 

0 

2 

2 

3 

3 

3 

3 

4 

4 

4 

5 

5 

5 

0 

6 

6 

7 

7 

8 

8 

126 

56 

0 

0 

2 

2 

2 

3    3 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

*? 

7 

8 

124 

58 

0 

0 

1 

2 

2 

2 

3 

3    3 

3 

4 

'4 

4 

v'S 

5 

6 

,6 

6 

6 

7 

7 

122 

60 

0 

0 

0 

2 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

5 

5 

5 

5 

6 

6 

7 

120 

62 

0 

0 

0 

2 

2 

2!   2 

2 

3 

3 

a 

3 

4 

4 

4 

5 

5 

5 

5 

6 

6 

118 

64 

0 

0 

0 

1 

2 

2    2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

4 

5 

5 

5 

6 

116 

66 

CO 

0 

0 

0 

2 

2     2 
20 

2 

2 

2 

3 

3 

3 

0 

3 

3" 

4 

4 

4 

4 

4 

5 

fi 

114 

bo 

70 

0 

0 
0 

0 
0 

0 

1 

1 

1 

2 
1     2 

::2 

2 

2 

2 
2 

3 
2 

0 

2 

3 

3 

3 

3 

3 

4 

4 

5 
4 

110 

74 

0 

0 

0 

0 

0 

1 

1 

1     1 

1 

1 

2 

2 

2i   2 

2 

2 

3 

3 

3 

3 

3 

3 

106 

78 

0 

0 

0 

0 

0 

0 

1 

1 

1 

l    1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

2 

2 

2 

102 

82 

0 

0 

0 

0 

0 

0 

0 

0 

1 

1 

1    1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

2 

2 

98 

86 

0 

0 

0 

0 

0 

0 

0 

0 

0    0 

0    0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

1 

94 

90° 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

90° 

Appar- 

i 

7 

id 

iil 

11 

16 

18 

20 

If 

22 

ii 

il 

25 

m 

2728 

B 

SO 

si  it 

ii 

34 

» 

36 

Appar- 
ent 

cnt 
Dis- 

Dis- 

tance. 

FIRST   CORRECTION  OF   DISTANCE. 

tance. 

36 


TABLE    VI. 

Second  Correction  of  the  Lunar  Distance. 


Appar- 

FIRST    CORRECTION   OF   DISTANCE.                        Appar. 

ent 

Tii^j 

ent 

Tlia 

ins- 
tance. 

H7 

58 

;ig' 

40 

41 

d 

43 

44 

4546 

4) 

48 

d 

ill 

ii 

§i 

§i 

4 

5> 

Si 

57 

58 

flf 

60' 

UlS- 

tance. 

Subtr 

,.         A,W 

15°  0' 

45 

47 

50 

52 

55 

57 

60 

63 

66 

69 

72 

75 

78 

81 

85 

88 

91 

95 

99 

102 

106 

no 

113 

117 

30 

43 

45 

48 

50 

53  56  58 

61 

64  67 

70 

72 

76 

79 

82 

85 

88  92 

95 

99 

102 

106 

110 

113 

16    0 

42|44 

40 

49 

51  54 

56 

59 

62  64 

67 

70 

73 

76 

79 

82 

85  89 

92 

95 

99 

102 

106 

110 

30 

40 

43 

45 

47 

50i52 

54 

57 

60162 

65 

68 

71 

74 

77 

80 

83  86 

89 

92 

96 

99 

103 

106 

17    0 

39 

4  1 

43 

46 

48;  50 

53 

55 

58 

60 

63 

66 

69 

71 

74 

77 

80  83 

86 

90 

93 

96 

99 

103 

30 

98 

40 

42 

44 

47|49 

51 

54 

56  59 

61 

64 

66 

69 

72 

75 

78 

81 

84 

87 

90 

93 

96 

100 

18    0 
30 

37 
36 

39 

38 

41 

40 

43 

42 

45147 
44  46 

50 
48 

52 

50 

54  57  59 
53  55  58 

62 
60 

64 
63 

67 
65 

70 
68 

73 
71 

75 
73 

78 
76 

81 

79 

84 

82 

87 
85 

90 

88 

94 
91 

97 

94 

19    0 

35 

37 

.-59 

41 

43 

45  47 

49 

51  54 

56 

58 

61 

63 

66 

69 

71 

74 

77 

79 

82 

85 

88 

91 

30 

34 

36 

.'57 

.'39 

41 

43  46 

48 

50 

52 

54 

57 

59 

62 

64 

67 

69 

72 

75 

77 

80 

83 

86 

89 

20 

33 

35 

36 

38 

40 

42  44 

4(5 

49 

51 

53 

55 

58 

60 

62 

65 

67 

70 

73 

75 

78 

81 

83 

86 

21 

31 

33 

.'55 

36 

38 

40  42 

44 

46 

48 

50 

52 

55 

57 

59 

61 

64 

66 

69 

71 

74 

76 

79 

82 

22 

30 

31 

33 

35 

36 

38  40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

61163 

65 

68 

70 

73 

75 

78 

23 

28 

30 

31 

33 

.'35 

36 

38 

40 

42 

44 

45 

47 

49 

51 

53 

56 

58  60 

62 

64 

67 

69 

72 

74 

24 

'27 

28 

.'50 

31 

33 

.'55 

3(1 

38 

40 

41 

43 

45 

47 

49 

51 

53 

55 

57 

59 

61 

64 

66 

68 

71 

25 

26 

27 

28 

30 

.'51 

33 

35 

3fi 

38 

40 

41 

43 

45 

47 

49  51 

53 

55 

57 

59 

61 

63 

65 

67 

26 

25 

26 

27 

29 

30 

32 

33 

35 

36 

38 

40 

41 

43 

45 

47  48 

50 

52 

54 

56 

58 

60 

62 

64 

27 

33 

25 

26 

27 

29  30 

32 

33 

35 

36 

38 

39 

41 

43 

45 

46 

48 

50 

52 

54 

56 

58 

60 

62 

28 

28 

24 

25 

26 

28 

29 

30 

32 

33 

35 

36 

38 

39 

41 

43  44 

46 

48 

50 

51 

53 

55 

57 

59 

29 

22 

2-'} 

24 

25 

26 

28 

29 

30 

32 

33 

35 

36 

38 

39 

41  43 

44 

46 

48 

49 

51 

53 

55 

57 

30 

21 

22 

2,'! 

24 

25 

27 

28 

29 

31 

32 

33 

35 

36 

38 

39141 

42 

44 

46 

47 

49 

51 

53 

54 

31 

20 

21 

22 

2.'i 

24 

26 

27 

28 

29 

31 

32 

33 

35 

36 

38  39 

41 

42 

44 

46 

47 

49 

51 

52 

32 

19 

20 

21 

22 

2:5 

25 

26 

27 

28 

30 

31 

32 

.31 

85 

36  38 

39 

41 

42 

44 

45 

47 

49 

50 

33 

18 

19 

20 

22 

23 

24 

25 

26 

27 

28 

30 

31 

32 

34 

35  36 

38 

39 

41 

42 

44 

45 

47 

48 

34 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

29 

30 

31 

32 

34 

35 

36 

38 

39 

41 

42 

44 

45 

47 

35 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

28 

29 

30 

31 

32 

34 

35 

36 

38 

39 

40 

42 

43 

45 

36 

If. 

17 

18 

19 

20 

21 

22 

2.5 

24 

25 

27 

28 

29 

30 

31  32 

34 

35 

36 

38 

39 

40 

42 

43 

37 

L6 

17 

18 

19 

19 

20 

21 

22 

23 

25 

26 

27 

28 

29 

3031 

33 

34 

35 

36 

38 

39 

40 

42 

38 

15 

16 

17 

18 

19  20 

21 

22 

23 

24 

25 

26 

27 

28 

29J30 

31 

33 

34 

35 

36 

38 

39 

40 

39 

15 

16 

16 

17 

15 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28  29 

30 

31 

33 

34 

35 

36 

38 

39 

40 

14 

15 

16 

17 

17 

18 

L9 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

33 

34 

35 

36 

37 

1403 

42 

I.') 

14 

15 

16 

If) 

17 

18 

19 

20 

21 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

33 

34 

35 

138 

44 

12 

13|l4 

14 

15 

16 

17 

17 

18 

19 

20  21 

22 

23 

24  24 

25 

26 

27 

-28 

29 

30 

31 

33 

136 

46 

12 

12J13 

13 

14 

15 

16 

16 

17 

18 

19 

19 

20 

21 

2223 

24 

25 

26 

26 

27 

28 

29 

30 

134    ! 

48 

11 

11 

12 

13 

13 

14 

15 

15 

16  17 

17 

18 

19 

20 

20:21 

22 

23 

24 

25 

26 

26 

27 

28 

132 

50 

10 

11 

11 

12 

12 

13 

14 

14 

15  16  16  17 

18 

18 

19  20 

21 

21 

22 

23 

24 

25 

25 

26 

ISO 

52 

9 

10 

10 

11 

11 

12 

13  13    14  Ullo  16 

16 

17 

18|18 

19 

20 

21 

21 

22 

23 

24 

25 

128- 

54 

9 

9 

10 

10 

11 

11 

12  12    13  13  14  15 

15  16 

16  17 

18 

18 

19 

20 

21 

21 

22 

23 

126 

56 

8 

9 

9 

9 

10 

10 

11 

11 

12,;  12  13!  14 

14 

15  15J16 

17 

17 

18 

18 

19 

20 

20 

21 

124 

58 

7 

S 

8 

9- 

9 

10 

10 

11 

11  12  12<13 

13 

14 

14  15 

15 

16 

16 

17 

18 

18 

19 

20 

122 

CO 

7 

7 

8 

8 

8 

9 

9 

10'i  10  11  ill   12 

12 

13 

13J14 

14 

15 

15 

16 

16 

17 

18 

18 

120 

62 

6 

-7 

7 

7 

8 

8 

9 

9 

9  10  10  11 

11 

12 

12!  13 

13 

14 

14 

15 

15 

16 

16 

17 

118 

64 

G 

6 

6 

7 

'7 

8 

8 

8 

9     9 

9  10 

10 

11  11J12 

12 

12 

13 

13 

14 

14 

15 

15 

116    i 

66 

5 

6 

0 

G 

7 

7 

7 

8 

8     8 

9 

9 

9 

10  10  11 

11 

11 

12 

12 

13 

13 

14 

14 

114     ] 

68 

5 

5 

5 

6 

6 

6 

7 

7 

7i    7 

8 

8 

8 

9     9  10 

10 

10 

11 

11 

11 

12 

12 

13 

112 

70 

*7  A 

4 

5 

5 

5 

5 

6 

6 

6 

6     I 

7    7 

6f» 

8 

8 

8 

9 

7 

9 

7 

9 

10 

10 

10 

11 

11 

11 

110 

1  HA 

/4 

78 

GO 

8 

9 

3 
') 

3 

3 

.) 

3 

2 

3 

.3 

.) 

4      4     4 
2      o     Q 

D 

4    4 

O        Q 

4 
3 

5 

7 
5 

5 
3 

| 

5 

5 

4" 

6 

6 

6 

8 
6 

6 

9 

7 

lUb 

102 

no 

O«S 

86 

1 

I 

1 

1 

1 

1 

I 

j 

I 

I 

O        O 

i;  ! 

1 

2 

2 

2 

. 
2 

2 

2      2 

2 

2 

2 

2 

«7O 

94 

90° 

0 

0 

0 

0 

0 

0 

0 

0 

0 

o 

o 

0 

0 

0 

0 

0 

0 

0 

0      0 

0 

0 

0 

0 

90° 

Appar- 

ent 

37 

^ 
$0 

ri 

id 

11 

42 

n 

44   45  46  47 

.18 

49 

50 

•51 

3l 

51  M 

55   56;  <§? 

$8 

ri 

60  Lppar 

Dis- 

I Dia- 

tance. 

FIRST    CORRECTION   OF   DISTANCE.                          Itance. 

37 


TABLE    VII. 


For  finding  the  Correction  of  the  Lunar  Distance  for  the  Contraction  of  the  Moon's  Semidiameter. 


!                      TABLE   VII.   A.     GIVING  THE   ARGUMENT   FOR   TABLE  VII.  B. 

Reduced 

APPARENT     ALTITUDE    OF     j). 

P  and  R 

i  .and  tv 

o 

- 

o 

o 

n 

o 

o 

o 

o 

c 

o 

o 

c 

o 

o 

o 

D 

r 

o      o 

of>. 

§     §la 

(i 

(i 

7i 

8 

9 

(ii 

10 

11 

12 

13 

11 

Li 

It) 

17 

18 

20 

25 

It) 

40 

§0 

41' 

65    56 

42 

63|  54 

47 

41 

43 

62    53 

46 

40 

35 

44 

60 

51 

45 

39 

34!  30 

27 

45 

58 

50 

43 

38 

33 

30 

26 

24 

21 

20 

46 

57 

49 

42 

37 

33    29 

26 

29 

21 

19 

17 

15 

47 

56 

48 

41 

36 

32    28 

25 

23 

20 

19 

17 

14 

12 

10 

48 

54 

46 

40 

35 

31    28 

25 

22 

20 

18 

17 

14 

12 

10 

9 

8 

7 

6 

49 

53 

45 

39 

35 

30   27 

24 

22 

19 

18 

16 

14 

12 

10 

9 

8 

7 

6 

6 

5 

3 

50 

52 

44 

38 

34 

30!  26 

24 

21 

19 

17 

16 

13 

11 

10 

9 

8 

7 

6 

5 

5 

3 

9 

2 

51 

50 

43 

38 

33 

29 

26 

23 

21 

19 

17 

15 

13 

11 

in 

.11 

7" 

7 

6 

5 

5 

3 

2 

2 

2 

52 

49 

42 

37 

32 

28  '  25 

23 

20 

18 

17 

15 

13 

11 

9 

8 

7 

7 

6 

5 

4 

3 

2 

2 

2 

53 

48 

41 

36 

32 

28    25 

22 

20 

18 

16 

15 

12 

11 

9 

8 

7 

C 

6 

5 

4 

3 

2 

2 

2 

54 

47 

41 

35 

31 

27    24 

22 

19 

18 

16 

15 

12 

10 

9 

8 

7 

6 

6 

5 

4 

3 

2 

2 

2 

55 

35 

30 

27 

24 

21 

19 

17 

16 

14 

12 

10 

9 

8 

7 

e 

G 

5 

4 

3 

2 

2 

2 

56 

26 

23 

21 

19 

17 

15 

14 

12 

10 

9 

8 

7 

i 

5 

5 

4 

3 

2 

2 

2 

57 

18 

17 

15 

14 

12 

10 

9 

7 

7 

t 

5 

5 

4 

3 

2 

2 

2 

58 

13 

11 

10 

8 

7 

7 

f 

5 

5 

4 

3 

2 

2 

2 

59 

8 

7 

6 

G 

5 

5 

4 

3 

2 

2 

2 

60 

4 

3 

2 

;    2 

2 

TABLE   VII.   B.    CONTRACTION  OF    ]>'S   SEMIDIAMETER. 

Wnole 

ARGUMENT  =  \UMBER    FROM    TABLE     VII.     A. 

Correction 
of  }>. 

2 

4 

6 

8 

10 

12 

M 

10 

13 

mm 

24 

21) 

2s 

30 

52 

11 

16 

IS 

40 

44 

48 

rl 

51) 

se 

64 

o' 

0 

0 

0 

0 

0 

0 

0 

0 

0 

G 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

5 

0 

0 

0 

0 

0 

0 

0 

0 

0 

"0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

10     - 

0 

0 

0 

0 

0 

0 

0 

0 

0 

c 

0 

0 

i 

1 

1 

1 

1 

1 

1 

1 

l 

1 

i 

1 

1 

1 

15 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

i 

1 

1 

1 

'2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

20 

0 

0 

0 

1 

1 

I 

1 

1 

1 

2 

2 

2 

2 

.2 

2 

3 

3 

3 

3 

3 

4 

4 

4 

4 

5 

5 

22 

d 

0 

1 

1 

1 

1 

I 

2 

2 

* 

2 

2 

3 

3 

3 

3 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

24 

0 

0 

1 

1 

I 

2 

2 

2.2 

3 

3 

3 

3 

3 

4 

4 

4 

4 

5 

5 

6 

6 

6 

7 

7 

26 

0 

1 

1 

1 

2 

2 

2 

2  a 

3 

3 

4 

4 

4 

4 

5 

5 

5 

5 

6 

6 

7 

8 

8 

9 

28 

0 

1 

1 

2 

2 

2 

3 

3 

a 

3 

4 

4 

4 

5 

5 

5 

6 

B 

6 

"7 

8 

8 

9 

9 

10 

30 

0 

t 

1 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

7 

7 

8 

9 

9 

10 

11 

12 

32 

0 

I 

2 

2 

2 

3 

3 

4 

4 

5 

5 

5 

6 

6 

7 

7 

7 

8 

8 

9 

10 

11 

11 

12 

13 

34 
36 

0 

1 
1 

2 

2 
2 

2 
3 

3 
3 

3 
4 

4 

4 

4 
5 

5 

5 

5 
6 

6 
6 

6 
7 

6 
7 

i 

8 

7 

8 

8 
•9 

8 
9 

9 
10 

9 

10 

10 
11 

11 
12 

12 
13 

13 
15 

14  15 

16  17 

38 

I 

2'   2 

3 

3 

4 

5 

5 

6 

6 

7 

8    8 

9 

9 

10 

10 

11 

12 

13 

14 

15 

16 

17  18 

40 

I 

2    3 

3 

4 

4 

5 

6 

1 

7 

8 

8    9 

9 

10 

ll 

12 

12 

13 

14 

15 

17 

18 

19  20 

42 

I 

2    3 

4 

4 

5 

6 

6 

$ 

8 

8 

9  10 

11 

ll 

12 

13 

13 

14 

16 

17 

18 

20 

21 

23 

44 

2 

2     3 

4 

5 

5 

6 

7    S 

9     9 

10  11   12!l2 

13 

14  15 

15 

17 

19 

20 

22 

23 

45 

2 

2!   3 

4 

5 

6 

6 

7    {• 

9J10 

11  11 

12  13 

14 

15 

15 

16 

18 

19 

21 

23 

24 

46 

2 

3     3 

4 

5 

6     7 

7    8 

9  10 

11  12 

13114 

14 

15 

16 

17 

19 

20 

22 

24 

47 

2 

3    4 

4 

5 

6 

•* 

8     9  10.11 

11   12 

13  14 

15 

16  17 

18 

19  21 

23 

25 

48 

2 

3    4 

5 

667 

8     9  10  ill 

12  13 

1415    16 

17  18 

18 

20  22!  24 

26 

49 

2 

3!  4 

5     6|   7     8 

9  10  11|12 

12  13 

14  15 

16 

17118 

19 

21  23 

25 

50 

2 

3     4 

5     6|   7     8 

13  14 

15 

16 

17 

18;19  20 

22  24 

26 

51 

2 

3!   4 

5 

6    7 

g 

9  10  1I!12 

14  15 

16 

17    18 

19  20 

21 

23  25 

27 

52 

2 

3     4 

5 

I    8 

q 

10  11J12  13 

14  15 

16 

17 

18  19  2l|22 

24i26 

53 

1 

2 

3 

4 

6 

7     8 

9 

10  11  12113 

15  16 

17 

18    19  20  21 

22 

25 

27 

54 

2 

3 

5 

6 

"I   8     9 

10  12:13114 

15  16 

17 

19    20  21 

22 

23 

26 

55 

2 

4 

5 

6 

7    8  10 

11  12  13  15 

16  17 

18 

19  >21  22 

56 

3 

4 

5 

6 

8 

9 

10 

ll  13  14  15 

16 

57 

1   4 

5 

7 

! 

—  ^^. 

When  the  nvartxt  limb  i*  observed  subtract  this  correction  ;  when  the  farthest,  add. 

TABLE    VIII. 

For  finding  the  Correr.tion  of  the  Lunar  Distance  for  the  Contraction  of  the  Sun's  Semidiameter. 


I                   TABLE  VIII.   A.     GIVING  THE   ARGUMENT   FOR  TABLE  VIII.  B. 

Reduced 

APPARENT    ALTITUDE    OF     0.                                                        || 

P.  and  R. 
of©. 

0 

§ 

o 
51 

6 

4 

7 

U 

§ 

0 

n 

n 

9 

0 

H 

10 

0 

11 

Q 

12 

0 

Ii 

o 

14 

if 

16 

17 

18 

10 

u 

10 

0 

40 

sol! 

1'    0" 

22 

18 

30 

30 

34 

24 

29 

2     0 

35 

37 

42 

46 

46 

30 

40 

42 

44 

47 

53 

59 

3     0 

44 

4G 

49 

51 

53 

57 

30 

45 

48 

51 

54 

57 

60 

62 

G7 

4     0 

45 

49 

52 

55 

59 

G2 

65 

63 

30 

47 

49 

51 

55 

59 

63 

66 

70 

5     0 

47 

50 

52 

54 

57 

Gl 

6G 

70 

74 

30 

47 

50 

52 

55 

57 

GO 

62 

67 

72 

6     0 

49 

52 

55 

57 

60 

G3 

6G 

68 

74 

30 

50 

53 

56 

59 

G2 

65 

G8 

71 

74 

7     0 

51 

54 

5S 

61    64 

07 

70 

74 

30 

55 

58J62 

65 

69 

72 

75 

8     0 

55 

59 

62 

56 

70 

73 

77 

30 

59 

63 

66 

70 

74 

78 

9     0 

6-2 

66 

70 

74 

79 

30 

66 

70 

74 

79 

10     0 

69 

74 

78 

30 

73 

77 

11     0 

76 

81 

30 

80 

TABLE  VIII.   B.    CONTRACTION  OF   0*8    SEMIDIAMETER. 

Whole 

ARGUMENT=  NUMBER    FROM    TABLE    VIII.    A.                                       | 

Correction 
of  Q. 

20 

i4 

28 

I! 

K 

40 

ii 

46 

48 

;5I) 

^ 

U 

§6 

58 

60 

61 

64 

66 

68 

70 

72 

74 

76 

78 

(      ii 
0     0 

0 

0 

0 

0 

0 

0 

0 

6 

0 

0 

0 

0 

0 

b 

0 

0 

0 

'o 

0 

It 

0 

// 

0 

II 

0 

tl 

0 

u 
0 

1     0 

1 

i 

i 

i 

i 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

2     0 

2 

2 

2 

2 

2 

2 

2 

i 

1 

I 

1 

i 

1 

1 

1 

1 

1 

1 

1 

1 

1 

i 

30 

3 

3 

3 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

1 

i 

3     0 

4 

4 

K 

4 

5 

3 
5 

3 

3 

3 

3 

3 

3 

3 

3 

2 

2 

2 

2 

2 

2 

2 

30 

9 

7 

f, 

g 

g 

g 

r. 

4     0 

20 

7 

•7 

7 

7 

G 

6 

6 

6 

5 

0 

5 

5 

5 

5 

5 

5 

4 

4 

40 

9 

8 

8 

8 

7 

7 

7 

7 

G 

6 

6 

6 

6 

5 

5 

5 

5 

5     0' 

10 

9 

11 

9 

1  0 

9 
10 

8 

8 

8 

8 

7 

7 

7 

7 

6 

6 

6 

6 

6 

20 
40 

12 

1  \  t 

12 

11 

11 

10 

10 

10 

9 

9 

9 

9 

8 

8 

8 

8 

r 

6     0 

13 

12 

12 

12 

11 

11 

10 

10 

10 

10 

9 

9 

9 

9 

8 

20 

14 

14 

13 

13 

12 

12 

12 

11 

11 

11  10 

10 

10 

10 

9 

40 

16 

15 

15 

14 

14 

13 

13 

13 

12 

12  11 

11 

11 

11 

10 

7     0 

18 

17 

16 

16 

15 

15 

14 

14 

13 

13  13 

12 

12 

12 

'1 

20 
40 

19 

IS 
20 

17 
19 

17 

18 

16 
18 

16 

17 

15 
17 

15 
16 

14.14 
16|l5 

13 
15 

13 
14 

13 
14 

i  > 
14 

8     0 

21 

21 

20 

19 

19 

18 

17 

17 

16 

1G 

16 

15 

15 

20 

22 

21 

20 

20 

19 

18 

18 

17 

17 

16 

16 

40 

23 

23 

22 

21 

20 

20 

19 

19 

18 

18 

17 

9     0 

24 

23 

22 

21 

21 

20 

20 

19 

19 

20 

25 

24 

23 

22 

22 

21 

21 

20 

40 

25 

25 

24 

23 

23 

22 

22 

10    0 

26 

2G 

25 

24 

24 

23 

20 

28 

27 

26 

25 

25 

40 

28 

28 

27 

26 

11     0 

29 

28 

20 

•. 

30 

Subtract  this  correction  fronri  the  distance. 

39 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

d 

• 

2 

• 

3 

1 

a 

5 

n 

6 

n 

7 

// 

8 

9 

O         |           II 

Oh.   Qm.  p. 

0.0000 

0.3010 

0.4771 

0.6021 

0.6990    0.7782 

0.8451 

0.9031 

0-9542 

0     10 

1.0000 

1.0414 

1.0792 

1.1139     1.1461 

1.1761     1.2041 

1.2304 

1.2553 

1.2788 

0    20 

1.3010 

1.3222 

1.3424 

1.3617     1.3802 

1.3979!    1.4150 

1.4314 

1.4472 

1.4624 

0    30 

1.4771 

1.4914     1.5051 

1.5185     1.5315 

1.5441     1.5563 

1.5682 

1.5798 

1.5911 

0    40 

1.6021 

1.6128 

1.6232 

1.6335     1.6435 

1.6532     1.6628 

1.6721 

.6812 

1.6902 

0    50 

1.6990 

1.7076 

1.7160 

1.7243 

1.7324 

1.7404     1.7482 

1.7559 

.7634 

1.7709 

0             0 

1.7782 

1.7853 

1.7924 

1.7993 

1.8062 

1.8129     1.8195 

1.8261 

.8325 

1.8388 

10 

1.8451 

1.8513 

1.8573 

1.8633 

1.8692 

1.8751 

1.8808 

1.8865 

.8921 

1.8976 

20 

1.9031 

1.9085 

1.9138 

1.9191 

1.9243 

1.9294 

1.9345 

1.9395 

.9445 

1.9494 

30 

1.9542 

1.9589 

1.9638 

1.9685 

1.9731 

1.9777     1.9823 

1.9868 

.9912 

1.9956 

40 

2.0000 

2.0043 

2.0086 

2.0128 

2.0170 

2.0212     2.0253 

2.0294 

2.0334 

2.0374 

50 

2.0414 

2.0453 

2.0492 

2.0531 

2.0569 

2.0607 

2.0645 

2.0682 

2.0719 

2.0755 

020 

2.0792 

2.0828 

2.0864 

2.0899 

2.0934 

2.0969 

2.1004 

2.1038 

2.1072 

2.1106 

2    10 

2.1139 

2.1173 

2.1206 

2.1239 

2.1271 

2.13031    2.1335 

2.1367 

2.1399 

2.1430 

2    20 

2.1461 

2.1492 

2.1523 

2.1553 

2.1584 

2.1614!    2.1644 

2.1673 

2.1703 

2.1732 

2    30 

2.1761 

2.1790 

2.1818 

2.1847 

2.1875 

2.1903 

2.1931 

2.1959 

2.19-87 

2.2014 

2    40 

2.2041 

2.2068 

2.2095 

2.2122 

2.2148 

2.2175 

2.2201 

2.2227 

2.2253 

2.2279 

2    50 

2.2304 

2.2330 

2.2355 

2.2380 

2.24.05 

2.2430 

2.2455 

2.2480 

2.2504 

2.2529 

030 

2.2553 

2.2577 

2.2601 

2.2625 

2.2648 

2.2672 

2.2695 

2.2718 

2.2742 

2.2765 

3    10 

2.2788 

2.2810 

2.2833 

2.2856 

2.2878 

2.2900 

2.2923 

2.2945 

2.2967 

2.2989 

3    20 

2.3010 

2.3032 

2.3054 

2.3075 

2.3096 

2.3118 

2.3139 

2.3160 

2.3181 

2.3201 

3    30 

2.3222 

2.3243 

2.3263 

2.3284 

2.3304 

2.3324 

2.3345 

2.3365 

2.3385!   2.3404 

3    40 

2.3424 

2.3444 

2.3464 

2.3483 

2.3502 

2.3522 

2.3541 

2.3560 

2.3579!    2.3598     • 

3    50 

23617 

2.3636 

2.3655 

2.3674 

2.3692 

2.3711 

2.3729 

2.3747 

2.3766     2.3784     ' 

040 

2.3802 

2.3820 

2.3838 

2.3856 

2.3874 

2.3892 

2.3909 

2.3927 

2.3945!   2.3962 

4    10 

2.3979 

2.3997 

2.4014 

2.4031 

2.4048 

2.4065 

2.4082 

2.4099 

2.4116i   2.4133 

4    20 

2.4150 

2.4166 

2.4183 

2.4200 

2.4216 

2.4232 

2.42491   2.4265 

2.4281!    2.4298 

4    30 

2.4314 

2.4330 

2.4346 

2.4362 

2.4378 

2.4393 

2.4409 

2.4425 

2.4440     2.4456 

4    40 

2.4472 

2.4487 

2.4502 

2.4518 

2.4533 

2.4548 

2.4564 

2.4579 

2.4594     2.4609 

4    50 

'2.4624 

2.4639 

2.4654 

2.4669 

2.4683 

2.4698 

2.4713 

2.4728 

2.4742 

2.4757 

050 

2.4771 

2.4786 

2.4800 

2.4814 

2.4829 

2.4843 

2.4857 

2.4871 

2.4886 

2.4900 

5    10 

2.4914 

2.4928 

2.4942 

2.4955 

2.4969 

2.4983 

2.4997 

2.5011 

2.5024 

2.5038 

5    20 

2.5051 

2.5065 

2.5079 

2.5092 

2.5105 

2.5119 

2.5132 

2.5145 

2.5159 

2.5172 

5    30 

2.5185 

2.5198 

2.5211 

2.5224 

2.5237 

2.5250 

2.5263 

2.5276 

2.5289 

2.5302 

5    40 

2.5315 

2.5328 

2.5340 

2.5353 

2.5366 

2.5378 

2.5391 

25403 

2.5416 

2.5428 

5    50 

2.5441 

2.5453 

2.5465 

2.5478 

2.5490 

2.5502 

2.5514 

2.5527 

2.5539 

2.5551 

060 

2.5563 

2.5575 

2.5587 

2.5599 

2.5611 

2.5623 

2.5635 

2.5647 

2.5658 

2.5670 

6    10 

2.5682 

2.5694 

2.5705 

2.5717 

2.5729 

2.5740 

2.5752 

2.5763 

2.5775 

2.5786 

6    20 

2.5798'   2.5809 

2.5821 

2.5832 

2.5843 

2.5855 

2.5866 

2.5877 

2.5888 

2.5899 

6    30 

2.5911 

2.5922 

2.5933 

2.5944 

2.5955 

2.5966 

2.5977 

2.5988 

2.5999 

2.6010 

6    40 

2.6021 

2.6031 

2.6042 

2.6053 

2.6064 

2.6075 

2.6085 

2.6096 

2.6107 

2.6117 

6    50 

2.6128 

2.6138 

2.6149 

2.6160 

2.6170 

2.6180 

2.6191 

2.6201 

2.6212 

2.6222 

070 

2.6232 

2.6243 

2.6253 

2.6263 

2.6274 

2.6284 

2.6294 

2.6304 

2.6314 

2.6325 

7    10 

2.6335 

2.6345 

2.6355 

2.6365 

2.6375 

2.6385 

2.6395 

2.6405 

2.6415 

2.6425 

7    20 

2.6435 

2.6444 

2.6454 

2.6464 

2.6474 

2.6484 

2.6493 

2.6503 

2.6513 

2.6522 

7    30 

2.6532 

2.6542 

2.6551 

2.6561 

2.6571 

2.6580 

2.6590 

2.6599 

2.6609 

2.6618 

7    40 

2.6628;    2.6637 

2.6646 

2.6656 

2.6665 

2.6675 

^.6684 

2.6693 

2.6702 

2.6712 

7    50 

2.6721     2.6730 

2.6739 

2.6749 

2.6758 

2.6767 

2.6776 

2.6785 

2.6794 

2.6803 

080 

2.6812^    2.6821 

2.6830 

2.68391    2.6848 

2.6857 

2.6866 

2.6875 

2.6884 

2.6893 

8    10 

2.6902     2.6911 

2.6920 

2.69281    2.6937 

2.6946 

2.6955 

2.6964 

2.6972 

2.6981 

8    20 

2.6990     2.6998 

2.7007 

2.7016 

2.7024 

2.7033 

2.7042 

2.7050 

2.7059 

2.7067 

8    30 

2.7076     2.7084 

2.7093 

2.7101 

2.7110 

2.7118 

2.7126 

2.7135 

2.7143 

2.7152 

8    40 

2.7160     2.7168 

2.7177 

2.7185 

2.7193 

2.7202 

2.7210 

2.7218 

2.7226 

2.7235 

8    50 

2.7243     2.7251 

2.7259 

2.7267 

2.7275 

2.7284 

2.7292 

2.7300 

2.7308 

2.7316 

090 

2.7324     2.7332 

2.7340 

2.7348 

2.7356 

2.7364 

2.7372 

2.7380 

2.7388 

2.7396 

9    10 

2.7404     2.7412 

2.7419 

2.7427 

2.7435 

2.7443 

2.7451 

2.7459 

2.7466 

2.7474 

9    20 

2.7482     2.7490 

2.7497 

2.7505 

2.7513 

2.7520 

2.7528 

2.7536 

2.7543 

2.7551 

9    30 

2.7559     2.7566 

2.7574 

2.7582     2.7589 

2.7597 

2.7604 

2.7612 

2.7619 

2.7627 

9    40 

2.7634     2.7642!    2.7649 

2.7657 

2.7664 

2.7672 

2.7679 

2.7686 

2.7694 

2.7701 

?    50 

2.7709     2.7716     2.7723 

2.7731 

2.7738 

2.7745 

2.7752 

2.7760 

2.7767 

2.7774 

TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

l^r. 

n 

0 

i 

2 

3 

n 

4 

5 

6 

7 

8 

n 

9 

Oh-lOm'  0'- 

2.7782 

2.7789 

2.7796,'  2.7803 

2.7810 

2.7818 

2.7825 

2.7832 

2.7839 

2.7846 

10  10 

2.7853 

2.7860 

2.7868|  2.7875 

2.7882 

2.7889 

2.7896 

2.7903 

2.7910 

2.7917 

10  20 

27924 

2.7931 

2.7938 

2.7945 

2.7952 

2.7959 

2.7966 

2.7973 

2.7980 

2.7987 

10  30 

2.7993 

2.8000 

2.8007 

2.8014 

2.8021 

2.8028 

2.8035 

2.8041 

2.8048 

2.8055 

10  40 

2.8062 

2.8069 

2.8075 

2.8082 

2.8089 

2.8096 

2.8102 

2.8109 

2.8116 

2.8122 

10  50 

2.8129 

2.8136 

2.8142 

2.8149 

2.8156 

2.8162 

2.8169 

2.8176 

2.8182 

2.8189 

0  11   0 

2.8195 

2.8202 

2.8209 

2.8215 

2.8222 

2.8228 

2.8235  2.8241 

2.8248 

2.8254 

11  10 

2.8261 

2.8267 

2.8274 

2.8280 

2.8287 

2.8293 

2.8299 

2.8306 

2.8312 

2.8319 

11  20 

2.8325 

2.8331 

2.8338 

2.8344 

2.8351 

2.8357 

2.8363 

2.8370 

2.8376 

2.8382 

11  30 

2.8388 

2.8395 

2.8401 

2.8407 

2.8414 

2.8420 

2.8426 

2.8432 

2.8439 

2.8445 

11  40 

2.8451 

2.8457 

2.8463 

2.8470 

2.8476 

2.8482 

2.8488 

2.8494 

2.8500 

2.8506 

11  50 

2.8513 

2.8519 

2.8525 

2.8531 

2.8537 

2.8543 

2.8549'  2.8555 

2.8561 

2.8567 

0  12   0 

2.8573 

2.8579 

2.8585 

2.8591 

2.8597 

2.8603 

2.8<>  '.I  i.8615 

2.8621 

2.8627 

12  10 

2.8633 

2.8639 

2.8645 

2.8651 

2.8657 

2.8663 

2.8tit:v)  2.8675 

2.8681 

2.8686 

12  20 

2.8692 

2.8698 

2.8704 

2.8710 

2.8716 

2.8722 

2.S7SS7.  2.8733 

2.8739 

2.8745 

12  30 

2.8751 

2.8756 

2.8762 

2.8768 

2.8774 

2.8779 

2.87  S.V  2.8791 

2.8797 

2.8802 

12  40 

2.8808 

2.8814 

2.8820 

2.8825 

2.8831 

2.8837 

2.8842 

2.8848 

2.8854 

2.8859 

12  50 

2.8865 

2.8871 

2.8876  2.8882 

2.8887 

2.8893 

2.8899 

2.8904 

2.8910 

2.8915 

0  13   0 

2.8921 

2.8927 

2.8932 

2.8938 

2.8943 

2.8949 

2.8954 

2.8960 

2.8965 

2.8971 

13  10 

2.8976 

2.8982 

2.8987 

2.8993 

2.8998 

2.9004 

2.9009 

2.9015 

2.9020  2.9025 

13  20 

2.9031 

2.9036 

2.9042 

2.9047 

2.9053 

2.9058 

2.9063 

2.9069 

2.9074;  2.9079 

13  30 

2.9085 

2.9090 

2.9096 

2.9101 

2.9106 

2.9112 

2.9117 

2.9122 

2.9128  2.9133 

13  40 

2.9138 

2.9143 

2.9149 

2.9154 

2.9159 

2.9165 

2.9170  2.9175 

2.9180  2.9186 

13  50 

2.9191 

2.9196 

2.9201 

2.9206 

2.9212 

2.9217 

2.9222 

2.9227 

2.9232 

29238 

0  14   0 

2.9243 

2.9248 

2.9253  2.9258 

2.9263 

2.9269 

2.9274 

2.9279 

2.9284 

2.9289 

14  10 

2.9294 

2.9299 

2.9304 

2.9309 

2.9315 

2.9320 

2.9325 

2.9330 

2.9335 

2.9340 

14  20 

2.9345 

2.9350 

2.9355 

2.9360 

2.9365 

2.9370 

2.9375 

2.9380 

29385  2.9390 

1    14  30 

2.9395 

2.9400 

2.9405 

2.9410 

2.9415 

2.9420 

2.9425 

2.9430 

2.9435  2.9440 

14  40 

2.9445 

2.9450 

2.9455 

2.9460 

2.9465 

2.9469 

2.9474 

2.9479 

2.9484  2.9489 

14  50 

2.9494 

2.9499 

2.9504 

2.9509 

2.9513 

2.9518 

2.9523 

2.9528 

2.9533  2.9538 

0  15   0 

2.9542 

2.9547 

2.9552 

2.9557 

2.9562 

2.9566 

2.9571 

2.9576 

2.9581  2.9586 

15  10 

2.9590 

2.9595 

2.9600 

2.9605 

2.9609 

2.9614 

2.9619 

2.9624 

2.9628  2.9633 

15  20 

2.9638 

2.9643 

2.9647 

2.9652 

2.9657 

T9661 

2.9666 

2.9671 

2.9675  2.9680 

15  30 

2.9685 

2.9689 

2.9694 

2.9699 

2.9703 

2.9708 

2.9713 

2.9717 

2.9722  2.9727 

15  40 

2.9731 

2.9736 

2.9741 

2.9745 

2.9750 

2.9754 

2.9759 

2.9763 

2.9768  2.9773 

15  50 

2.9777 

2.9782 

2.9786 

2.9791 

2.9795 

2.9800 

2.9805 

--2^809 

2.9814  2.9818 

0  16   0 

2.9823 

2.9827 

2.9832 

2.9836 

2.9841 

2.9845 

2.9850 

2.9854 

2.9859  2.9863 

16  10 

2.9868 

2.9872 

29877 

2.9881 

2.9886 

2.9890 

2.9894 

2.9899 

2.9903  2.9908 

16  20 

2.9912 

2.9917 

2.9921 

2.9926 

2.9930 

2.9934 

2.9939 

2.9943 

2.9948  2.9952 

16  30 

2.9956 

2.9961  2.9965 

2.9969 

2.9974 

2.9978 

2.9983 

2.9987 

2.9991  2.9996 

16  40 

3.0000 

3.0004  3.0009 

3.0013 

3.0017 

3.0022 

3.0026 

3.0030 

3.0035  3.0039 

16  50 

3.0043 

3.0048  3.0052 

3.0056 

3.0060 

3.0065 

3.0069 

3.0073 

3.0077  3.0082 

0  17   0 

3.0086 

3.0090  3.0095 

3.0099 

3.0103 

3.0107 

3.0111 

3.0116 

3.0120  3.0124 

17  10 

3.0128 

3.0133  3.0137 

3.0141 

3.0145 

3.0149 

3.0154 

3.0158 

3.0162  3.0166 

17  20 

3.0170 

3.0175  3.0179 

3.0183 

3.0187 

3.0191 

3.0195 

3.0199 

3.0204'  3.0208 

17  30 

3.0212 

3.0216  3.0220 

3.0224 

3.0228 

3.0233 

3.0237 

3.0241 

3.0245  3.0249 

17  40 

3.0253 

3.0257  3.0261 

3.0265 

3.0269 

3.0273 

3.0278 

3.0282 

3.0286  '  3.0290 

17  50 

3.0294 

3.0298  3.0302 

3.0306 

3.0310 

3.0314 

3.0318 

3.0322 

3.0326  \  3.0330 

0  18   0 

3.0334 

3.0338  ;  3.0342 

3.0346 

3.0350 

3.0354 

3.0358 

3.0362 

3.0366  3.0370 

18  10 

3.0374 

3.0378'  3.0382 

3.0386 

3.0390 

3.0394 

30398 

3.0402 

3.0406  3.0410 

18  20 

3.0414  3.041  8  1  3.0422 

3.0426 

3.0430 

3.0434 

3.0438 

3.0441 

3.0445 

3.0449 

18  30 

3.0453  3.0457  3.0461 

3.0465 

3.0469 

3.0473 

3.0477 

3.0481 

3.0484 

3.0488 

18  40 

3.0492  3.04961  3.0500 

3.0504 

3.0508 

3.0512 

3.0515 

3.0519 

3.0523 

3.0527 

18  50 

3.0531  3.0535  3.0538 

3.0542 

3.0546 

3.0550 

3.0554 

30558 

3.0561 

3.0565 

0  19   0 

3.0569  3.0573  3.0577 

3.0580 

3.0584 

3.0588 

3.0592 

3.0596 

3.0599 

3.0603 

19  10 

3.0607  3.0611 

3.0615 

3.0618 

3.0622 

3.0626 

3.0630 

3.0633 

3.0637 

3.0641 

19  20 

3.0645  !  3.0648 

3.0652!  3.0656 

3.0660 

3.0663 

3.0667 

3.0671 

3.0674 

3.0678 

19  30 

3.0682'  3.0686 

3.0689  3.0693 

3.0697 

3.0700 

3.0704 

3.0708 

3.0711 

3.0715 

19  40 

3.0719  3.0722 

3.0726  3.0730 

3.0734 

3.0737 

3.0741 

3.0745  3.0748 

3.0752 

19  50 

3.0755  3.0759 

3.0763 

3.0766 

3.0770 

3.0774 

3.0777 

3.0781 

3.0785 

3.0788 

TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

n 

0 

i 

2 

n 

3 

• 

4 

5 

6 

n 
1 

n 

8 

9 

0,          I          II 

Oh.20m.  QS. 

3.0792 

3.0795 

3.0799 

3.0803 

3.0806 

3.0810 

3.0813 

3.0817 

3.0821 

3.0824 

20    10 

3.0828 

3.0831 

3.0835 

3.0839 

3.0842 

3.0846 

3.0849 

3.0853 

3.0856 

3.0860 

20    20 

3.0864 

3.0867 

3.0871 

3.0874 

3.0878 

3.0881 

3.0885 

3.0888 

3.0892 

3.0896 

20    30 

3.0899 

3.0903 

3.0906 

3.0910 

3.0913 

3.0917 

3.0920 

3.0924 

3.0927 

3.0931 

20    40 

3.0934 

3.0938 

3.0941 

3.0945 

3.0948 

3.0952 

3.0955 

3.0959 

3.0962 

3.0966 

20    50 

3.0969 

3.0973 

3.0976 

3.0980 

3.0983 

3.0986 

3.0990 

3.0993 

3.0997 

3.1000 

0   21      0 

3.1004 

3.1007 

3.1011 

3.1014 

3.1017 

3.1021 

3.1024 

3.1028 

3.1031 

3.1035 

21     10 

3.1038 

3.1041 

3.1045 

3.1048 

3.1052 

3.1055 

3.1059 

3.1062 

3.1065 

3.1069 

21    20 

3.1072 

3.1075 

3.1079 

3.1082 

3.1086 

3.1089 

3.1092 

3.1096 

3.1099 

3.1103 

21    30 

3.1106 

3.1109 

3.1113 

3.1116 

3.1119 

3.1123 

3.1126 

3.1129 

3.1133 

3.1136 

21    40 

3.1139 

3.1143 

3.1146 

3.1149 

3.1153 

3.1156 

3.1159 

3.1163 

3.1166 

3.1169 

21    50 

3.1173 

3.1176 

3.1179 

31183 

3.1186 

3.1189 

3.1193 

3.1196 

3.1199 

3.1202 

0  22      0 

3.1206 

3.1209 

3.1212 

3.1216 

3.1219 

3.1222 

3.1225 

3.1229 

3.1232 

3.1235 

22    10 

3.1239 

3.1242 

3.1245 

3.1248 

3.1252 

3.1255 

3.1258 

3.1261 

3.1265 

3.1268 

22    20 

3.1271 

3.1274 

3.1278 

3.1281 

3.1284 

3.1287 

3.1290 

3.1294 

3.1297 

3.1300 

22    30 

3.1303 

3.1307 

3.1310 

3.1313 

3.1316 

3.1319 

3.1323 

3.1326 

3.1329 

3.1332 

22    40 

3.1335 

3.1339 

3.1342 

3.1345 

3.1348 

3.1351 

3.1355 

3.1358 

3.1361 

3.1364 

22    50 

3.1367 

3.1370 

3.1374 

3.1377 

3.1380 

3.1383 

3.1386 

3.1389 

3.1392 

3.1396 

0   23      0 

3.1399 

3.1402 

3.1405 

3.1408 

3.1411 

3.1414 

3.1418 

31421 

3.1424 

3.1427 

23    10 

3.1430 

3.1433 

3.1436 

3.1440 

3.1443 

3.1446 

3.1449 

3.1452 

3.1455 

3.1458 

23    20 

3.1461 

3.1464 

3.1467 

3.1471 

3.1474 

3.1477 

3.1480 

3.1483 

3.1486 

3.1489 

23    30 

3.1492 

3.1495 

3.1498 

3.1501 

3.1504 

3.1508 

3.1511 

3.1514 

3.1517 

3.1520 

23    40 

3.1523 

3.1526 

3.1529 

3.1532 

3.1535 

3.1538 

3.1541 

3.1544 

3.1547 

3.1550 

23    50 

3.1553 

3.1556 

3.1559 

3.1562 

3.1565 

3.1569 

3.1572 

3.1575 

3.1578 

3.1581 

0   24      0 

3.1584 

3.1587 

3.1590 

3.1593 

3.1596 

3.1599 

3.1602 

3.1605 

3.1608 

3.1611 

24    10 

3.1614 

3.1617 

3.1620 

3.1623 

3.1626 

3.1629 

3.1632 

3.1635 

3.1638 

3.1641 

24    20 

3.1644 

3.1647 

3.1649 

3.1652 

3.1655 

3.1658 

3.1661 

3.1664 

3.1667 

3.1670 

24    30 

3.1673 

3.1676 

3.1679 

3.1682 

3.1685 

3.1688 

3.1691 

3.1694 

3.1697 

3.1700 

24    40 

3.1703 

3.1706 

3.1708 

3.1711 

3.1714 

3.1717 

3.1720 

3.1723 

3.1726 

3.1729 

24    50 

3.1732 

3.1735 

3.1738 

3.1741 

3.1744 

3.1746 

3.1749 

3.1752 

3.1755 

3.1758 

0  25      0 

3.1761 

3.1764 

3.1767 

3.1770 

3.1772 

3.1775 

3.1778 

3.1781 

3.1784 

3.1787 

25    10 

3.1790 

3.1793 

3.1796 

3.1798 

3.1801 

3.1804 

3.1807 

3.1810 

3.1813 

3.1816 

25    20 

3.1818 

3.1821 

3.1824 

3.1827 

3.1830 

3.1833 

3.1836 

3.1838 

3.1841 

3.1844 

25    30 

3.1847 

3.1850 

3.1853 

3.1855 

3.1858 

3.1861 

3.1864 

3.1867 

3.1870 

3.1872 

25    40 

3.1875 

3.1878 

3.1881 

3.1884 

3.1886 

3.1889 

3.1892 

3.1895 

3.1898 

3.1901 

25    50 

3.1903 

3.1906 

3.1909 

3.1912 

3.1915 

3.1917 

3.1920 

3.1923 

3.1926 

3.1928 

0  26      0 

3.1931 

3.1934 

3.1937 

3.1940 

3.1942 

3.1945 

3.1948 

3.1951 

3.1953 

3.1956 

26    10 

3.1959 

3.19G2 

3.1965 

3.1967 

3.1970 

3.1973 

3.1976 

3.1978 

3.1981 

3.1984 

26    20 

3.1987 

3.1989 

3.1992 

3.1995 

3.1998 

3.2000 

3.2003 

3.2006 

3.2009 

3.201  1 

26    30 

3.2014 

3.2017 

3.2019 

3.2022 

3.2025 

3.2028 

3.2030 

3.2033 

3.2036 

3.2038 

26    40 

3.2041 

3.2044 

3.2047 

3.2049 

3.2052 

3.2055 

3.2057 

3.2060 

3.2063 

3.2066 

26    50 

3.2068 

3.2071 

3.2074 

3.2076 

3.2079 

3.2082 

3.2084 

3.2087 

3.2090 

3.2092 

0  27      0 

3.2095 

3.2098 

3.2101 

3.2103 

3.2106 

3.2109 

3.2111 

3.2114 

3.2117 

3.2119 

27    10 

3.2122 

3.2125 

3.2127 

3.2130 

3.2133 

3.2135 

3.2138 

3.2140 

3.2143 

3.2146 

27    20 

3.2148 

3.2151 

3.2154 

3.2156 

3.2159 

3.2162 

3.2164 

3.2167 

3.2170 

3.2172 

27    30 

3.2175 

3.2177 

3.2180 

3.2183 

3.2185 

3.2188 

3.2191 

3.2193 

3.2196 

3.2198 

27    40 

3.2201 

3.2204 

3.2206 

3.2209 

3.2212 

3.2214 

3.2217 

3.2219 

3.2222 

3.2225 

27    50 

3.2227 

3.2230 

3.2232 

3.2235 

3.2238 

3.2240 

3.2243 

3.2245 

3.2248 

3.2250 

0  28      0 

3.2253 

3.2256 

3.2258 

3.2261 

3.2263 

3.2266 

3.2269 

3.2271 

3.2274 

3.2276 

28    10 

3.2279 

3.2281 

3.2284 

3.2287 

3.2289 

3.2292 

3.2294 

3.2297 

3.2299 

3.2302 

28    20 

3.2304 

3.2307 

3.2310 

3.2312 

3.2315 

3.2317 

3.2320 

3.2322 

3.2325 

3.2327 

28    30 

3.2330 

3.2333 

3.2335 

3.2338 

3.2340 

3.2343 

3.2345 

3.2348 

3.2350 

3.2353 

28    40 

3.2355 

3.2358 

3.2360 

3.2363 

3.2365 

3.2368 

3.2370 

3.2373 

3.2375 

3.2378 

28    50 

3.2380 

3.2383 

3.2385 

3.2388 

3.2390 

3.2393 

3.2395 

3.2398 

3.2400 

3.2403 

0   29      0 

3.2405 

3.2408 

3.2410 

3.2413 

3.2415 

3.2418 

3.2420 

3.2423 

3.2425 

3.2428 

29    10 

3.2430 

3.2433 

3.2435 

3.2438 

3.2440 

3.2443 

3.2445 

3.2448 

3.2450 

3.2453 

29    20 

3.2455 

3.2458 

3.2460 

3.24631   3.2465 

3.2467 

3.2470 

3.2472 

3.2475 

3.2477 

29    30 

3.2480 

3.2482 

3.2485 

3.2487 

3.2490 

3.2492 

3.2494 

3.2497 

3.2499 

3.2502 

29    40 

3.2504 

3.2507 

3.2509 

3.2512 

3.2514 

3.2516 

3.2519     3.2521 

3.2524 

3.2526 

29    50 

3.2529 

3.2531 

3.2533  j    3.2536    3.2538 

3.2541 

3.2543  1    3.2545     3.2548 

3.2550 

TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

d 

n 

2 

n 

3 

I 

5 

6 

7 

£ 

9 

O,    1    II 

Oh.30m.  QS. 

3.2553 

3.2555 

3.2558 

3.2560 

3.2562 

3.2565 

3.2567 

3.2570 

3.2572 

3.2574 

30  10 

3.2577 

3.2579 

3.2582 

3.2584 

3.2586 

3.2589 

3.2591 

3.2594 

3.2596 

3.2598 

30  20 

3.2601 

3.2603 

3.2605 

3.2608 

3.2610 

3.2613 

3.2615 

3.2617 

3.2620 

3.2622 

30  30 

3.2625 

3.2627 

3.2629 

3.2632 

3.2634 

3.2636 

3.2639 

3.2641 

3.2643 

3.2646 

30  40 

3.2648 

3.2651 

3.2653 

3.2655 

3.2658 

3.2660 

3.2662 

3.2665 

3.2667 

3.2669 

30  50 

3.2672 

3.2674 

3.2676 

3.2679 

3.2681 

3.2683 

3.2686 

3.2688 

3.2690 

3.2693 

0  31   0 

3.2695 

3.2697 

3.2700 

3.2702 

3.2704 

3.2707 

3.2709 

3.2711 

3.2714 

3.2716 

31  10 

3.2718 

3.2721 

3.2723 

3.2725 

3.2728 

3.2730 

3.2732 

3.2735 

3.2737 

3.2739 

31  20 

3.2742 

3.2744 

3.2746 

3.2749 

3.2751 

3.2753 

3.2755 

3.2758 

3.2760 

3.2762 

31  30 

3.2765 

3.2767 

3.2769 

3.2772 

3.2774 

3.2776 

3.2778 

3.2781 

3.2783 

3.2785 

31  40 

3.2788 

3.2790 

3.2792 

3.2794 

3.2797 

3.2799 

3.2801 

3.2804 

3.2806 

3.2808 

31  50 

3.2810 

3.2813 

3.2815 

3.2817 

3.2819 

3.2822 

3.2824 

3.2826 

3.2828 

3.2831 

0  32   0 

3.2833 

3.2835 

3.2838 

3.2840 

3.2842 

3.2844 

3.2847 

3.2849 

3.2851 

3.2853 

32  10 

3.2856 

3.2858 

3.2860 

3.2862 

3.2865 

3.2867 

3.2869 

3.2871 

3.2874 

3.2876 

32  20 

3.2878 

3.2880 

3.2882 

3.2885 

3.2887 

3.2889 

3.2891 

3.2894 

3.2896 

3.2898 

32  30 

3.2900 

3.2903 

3.2905 

3.2907 

3.2909 

3.2911 

3.2914 

3.2916 

3.2918 

3.2920 

32  40 

3.2923 

3.2925 

3.2927 

3.2929 

3.2931 

3.2934 

3.2936 

3.2938 

3.2940 

3.2942 

32  50 

3.2945 

3.2947 

3.2949 

3.2951 

3.2953 

3.2956 

3.2958 

3.2960 

3.2962 

3.2964 

0  33   0 

3.2967 

3.2969 

3.2971 

3.2973 

3.2975 

3.2978 

3.2980 

3.2982 

3.2984 

3.2986 

33  10 

3.2989 

3.2991 

3.2993 

3.2995 

3.2997 

3.2999 

3.3002 

3.3004 

3.3006 

3.3008 

33  20 

3.3010 

3.3012 

3.3015 

3.3017 

3.3019 

3.3021 

3.3023 

3.3025 

3.3028 

3.3030 

33  30 

3.3032 

3.3034 

3.3036 

3.3038 

3.3041 

3.3043 

3.3045 

3.3047 

3.3049 

3.3051 

3?  40 

3.3054 

3.3056 

3.3058 

3.3060 

3.3062 

3.3064 

3.3066 

3.3069 

3.3071 

3.3073 

33  50 

3.3075 

33077 

3.3079 

3.3081 

3.3084 

3.3086 

3.3088 

3.3090 

3.3092 

3.3094 

0  34   0 

3.3096 

3.3098 

3.3101 

3.3103 

3.3105 

3.3107 

3.3109 

3.3111 

3.3113 

3.3115 

34  10 

3.3118 

3.3120 

3.3122 

3.3124 

3.3126 

3.3128 

3.3130 

3.3132 

3.3134 

3.3137 

34  20 

3.3139 

3.3141 

3.3143 

3.3145 

3.3147 

3.3149 

3.3151 

3.3153 

3.3156 

3-3158 

34  30 

3.3160 

3.3162 

3.3164 

3.3166 

3.3168 

3.3170 

3.3172 

3.3174 

3.3176 

3.3179 

34  40 

3.3181 

3.3183 

3.3185 

3.3187 

3.3189 

3.3191 

3.3193 

3.3195 

3.3197 

3.3199 

34  50 

3.3201 

3.3204 

3.3206 

3.3208 

3.3210 

3.3212 

3.3214 

3.3216 

3.3218 

3.3220 

0  35   0 

3.3222 

3.3224 

3.3226 

3.3228 

3.3230 

3.3233 

3.3235 

3.3237 

3.3239 

3.3241 

35  10 

3.3243 

3.3245 

3.3247 

3.3249 

3.3251 

3.3253 

3.3255 

3.3257 

3.3259 

3.3261 

35  20 

3.3263 

3.3265 

3.3267 

3.3269 

3.3272 

3.3274 

3.3276 

3.3278 

3.3280 

3.3282 

35  30 

3.3284 

3.3286 

3.3288 

3.3290 

3.3292 

3  3294 

3.3296 

3.3298 

3.3300 

3.3302 

35  40 

3.3304 

3.3306 

3.3308 

3.3310 

3.3312 

3.3314 

3.3316 

3.3318 

3.3320 

3.3322 

35  50 

3.3324 

3.3326 

3.3328 

3.3330 

3.3332 

3.3334 

3.3336 

3.3339 

3.3341 

3.3343 

0  36   0 

3.3345 

3.3347 

3.3349 

3.3351 

3.3353 

3.3355 

3.3357 

3.3359 

3.3361 

3.3363 

36  10 

3.3365 

3.3367 

3.3369 

3.3371 

3.3373 

3.3375 

3.3377 

3.3379 

3.3381 

3.3383 

36  20 

3.3385 

3.3387 

3.3389 

3.3391 

3.3393 

3.3395 

3.3397 

3.3398 

3.3400 

3.3402 

36  30 

3.3404 

3.3406 

3.3408 

3.3410 

3.3412 

3.3414 

3.3416 

3.3418 

3.3420 

3.3422 

36  40 

3.3424 

3.3426 

3*3428 

3.3430 

3.3432 

3.3434 

3.3436 

3.3438 

3.3440 

3.3442 

36  50 

3.3444 

3.3446 

3.3448 

3.3450 

3.3452 

3.3454 

3.3456 

3.3458 

3.3460 

3.3462 

0  37   0 

3.3464 

3.3465 

3.3467 

3.3469 

3.3471 

3.3473 

3.3475 

3.3477 

3.3479 

3.3481 

37  10 

3.3483 

3.3485 

3.3487 

3.3489 

3.3491 

3.3493 

3.3495 

3.3497 

3.3499 

3.3501 

37  20 

3.3502 

3.3504 

3.3506 

3.3508 

3.3510 

3.3512 

3.3514 

3.3516 

3.3518 

3.3520 

37  30 

3.3522 

33524 

3.3526 

3.3528 

3.3530 

3.3531 

3.3533 

3.3535 

3.3537 

3.3539 

37  40 

3.3541 

3.3543 

3.3545 

3.3547 

3.3549 

3.3551 

3.3553 

3.3555 

3.3556 

3.3558 

37  50 

3.3560 

33562 

3.3564 

3.3566 

3.3568 

3.3570 

3.3572 

3.3574 

3.3576 

3.3577 

0  38   0 

3.3579 

3.3581 

3.3583 

3.3585 

3.3587 

3.3589 

3.3591 

3.3593 

3.3595 

3.3596 

38  10 

3.3598 

3.3600 

3.3602 

3.3604 

3.3606 

3.3608 

3.3610 

3.3612 

3.3614 

3.3615 

38  20 

3.3617 

3.3619 

3.3621 

3.3623 

3.3625 

3.3627 

3.3629 

3.3630 

3.3632 

3.3634 

38  30 

3.3636 

3.3638 

3.3640 

3.3642 

3.3644 

3.3646 

3.3647 

3.3649 

3.3651 

3.3653 

38  40 

3.3655 

3.3657 

3.3659 

3.3660 

3.3662 

3.3664 

3.3666 

3.3668 

3.3670 

3.3672 

38  50 

3.3674 

3.3675 

3.3677 

3.3679 

3.3681 

3.3683 

3.3685 

3.3687 

3.3688 

3.3690 

0  39   0 

3.3692 

3.3694 

3.3696 

3.3698 

3.3700 

3.3701 

3.3703 

3.3705 

3.3707 

3.3709 

39  10 

3.3711 

3.3713 

3.3714 

3.3716 

3.3718 

3.3720 

3.3722 

3.3724 

3.3725 

3.3727 

39  20 

3.3729 

3.3731 

3.3733 

3.3735 

3.3736 

3.3738 

3.3740 

3.3742 

3.3744 

3.3746 

39  30 

3.3747 

3.3749 

3.3751 

3.3753 

3.3755 

3.3757 

3.3758 

3.3760 

3.3762 

3.3764 

39  40 

3.3766 

3.3768 

3.3769 

3.3771 

3.3773 

3.3775  3.3777 

3.3779 

3.3780 

3.3782 

39  50 

3.3784 

3.3786 

3.3788 

3.3789 

3.3791 

3.3793  3.3795 

3.3797 

3.3798 

3.3800 

TABLE 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

d 

n 

2 

n 

3 

1 

a 

5 

6 

n 

7 

n 

8 

• 

9 

Oh-40m-  o',. 

3.3802 

3.3804 

3.3806 

3.3808 

3.3809 

3.3811 

3.3813 

3.3815 

3.3817 

3.3818 

40  10 

3.3820 

3.3822 

3.3824 

3.3826 

3.3827 

3.3829 

3.3831 

3.3833 

3.3835 

3.3836 

40  20 

3.3838 

3.3840 

3.3842 

3.3844 

3.3845 

3.3847 

3.3849 

3.3851 

3.3852 

3.3854 

40  30 

3.3856 

3.3858 

3.3860 

3.3861 

3.3863 

3.3865 

3.3867 

3.3869 

3.3870 

3.3872 

40  40 

3.3874 

3.3876 

3.3877 

3.3879 

3.3881 

3.3883 

3.3885 

3.3886 

3.3888 

3.3890 

40  50 

3.3892 

3.3893 

3.3895 

3.3897 

3.3899 

3.3901 

3.3902 

3.3904 

3.3906 

3.3908 

-  41   0 

3.3909 

3.3911 

3.3913 

3.3915 

3.3916 

3.3918 

3.3920 

3.3922 

3.3923 

3.3925 

41  10 

3.3927 

3.3929 

3.3930 

3.3932 

3.3934 

3.3936 

3.3938 

3.3939 

3.3941 

3.3943 

41  20 

3.3945 

3.3946 

3.3948 

3.3950 

3.3952 

3.3953 

3.3955 

3.3957 

3.3959 

3.3960 

41  30 

3.3962 

3.3964 

3.3965 

3.3967 

3.3969 

3.3971 

3.3972 

3.3974 

3.3976 

3.3978 

41  40 

3.3979 

3.3981 

3.3983 

3.3985 

3.3986 

3.3988 

3.3990 

3.3992 

3.3993 

3.3995 

41  50 

3.3997 

3.3998 

3.4000 

3.4002 

3.4004 

3.4005 

3.4007 

3.4009 

3.4011 

3.4012 

0  42   0 

3.4014 

3.4016 

3.4017 

3.4019 

3.4021 

3.4023 

3.4024 

3.4026 

3.4028 

3.4029 

42  10 

3.4031 

3.4033 

3.4035 

3.4036 

3.4038 

3.4040 

3.4041 

3.4043 

3.4045 

3.4047 

42  20 

3.4048 

3.4050 

3.4052 

3.4053 

3.4055 

3.4057 

3.4059 

3.4060 

3.4062 

3.4064 

42  30 

3.4065 

3.4067 

3.4069 

3.4071 

3.4072 

3.4074 

3.4076 

3.4077 

3.4079 

3.4081 

42  40 

3.4082 

3.4084 

3.4086 

3.4087 

3.4089 

3.4091 

3.4093 

3.4094 

3.4096 

3.4098 

42  50 

3.4099 

3.4101 

3.4103 

3.4104 

3.4106 

3.4108 

3.4109 

3.4111 

3.4113 

3.4115 

0  43   0 

3.4116 

3.4118 

3.4120 

3.4121 

3.4123 

3.4125 

3.4126 

3.4128 

3.4130 

3.4131 

43  10 

3.4133 

3.4135 

3.4136 

3.4138 

3.4140 

3.4141 

3.4143 

3.4145 

3.4146 

3.4148 

43  20 

3.4150 

3.4151 

3.4153 

3.4155 

3.4156 

3.4158 

3.4160 

3.4161 

3.4163 

3.4165 

43  30 

3.4166 

3.4168 

3.4170 

3.4171 

3.4173 

3.4175 

3.4176 

3.4178 

3.4180 

3.4181 

43  40 

34183 

3.4185 

3.4186 

3.4188 

3.4190 

3.4191 

3.4193 

3.4195 

3.4196 

3.4198 

43  50 

3.4200 

3.4201 

3.4203 

3.4205 

3.4206 

3.4208 

3.4209 

3.4211 

3.4213 

3.4214 

0  44   0 

3.4216 

3.4218 

3.4219 

3.4221 

3.4223 

3.4224 

3.4226 

3.4228 

3.4229 

3.4231 

44  10 

3.4232 

3.4234 

3.4236 

3.4237 

3.4239 

3.4241 

3.4242 

3.4244 

3.4246 

3.4247 

44  20 

3.4249 

3.4250 

3.4252 

3.4254 

3.4255 

3.4257 

3.4259 

3.4260 

3.4262 

3.4263 

44  30 

3.4265 

3.4267 

3.4268 

3.4270 

3.4272 

3.4273 

3.4275 

3.4276 

J.4278 

3.4280 

44  40 

3.4281 

3.4283 

3.4285 

3.4286 

3.4288 

3.4289 

3.4291 

3.4293 

3.4294 

3.4296 

44  50' 

3.4298 

3.4299 

3.4301 

3.4302 

3.4304 

3.4306 

3.4307 

3.4309 

3.4310 

3.4312 

0  45   0 

3.4314 

3.4315 

3.4317 

3.4318 

3.4320 

3.4322 

3.4323 

3.4325 

3.4326 

3.4328 

45  10 

3.4330 

3.4331 

3.4333 

3.4334 

3.4336 

3.4338 

3.4339 

3.4341 

3.4342 

3.4344 

45  20 

3.4346 

3.4347 

3.4349 

3.4350 

3.4352 

3.4354 

3.4355 

3.4357 

3.4358 

3.4360 

45  30 

3.4362 

3.4363 

3.4365 

3.4366 

3.4368 

3.4370 

3.4371 

3.4373 

3.4374 

3.4376 

45  40 

3.4378 

3.4379 

3.4381 

3.4382 

3.4384 

3.4385 

3.4387 

3.4389 

3.4390 

3.4392 

45  50 

3.4393  3.4395 

3.4396 

3.4398 

3.4400 

3.4401 

3.4403 

3.4404 

3.4406 

3.4408 

0  46   0 

3.4409  3.4411 

3.4412 

3.4414 

3.4415 

3.4417 

3.4419 

3.4420 

3.4422 

3.4423 

46  10 

3.4425!  3.4426 

3.4428 

3.4429 

3.4431 

3.4433 

3.4434 

3.4436 

3.4437 

3.4439 

46  20 

3.4440 

3.4442 

3.4444 

3.4445 

3.4447 

3.4448 

3.4450 

3.4451 

3.4453 

3.4454 

46  30 

3.4456 

3.4458 

3.4459 

3.4461 

3.4482 

3.4464 

3.4465 

3.4467 

3.4468 

3.4470 

46  40 

3.4472 

3.4473 

3.4475 

3.4476 

3.4478 

3.4479 

3.4481 

3.4482 

3.4484 

3.4486 

46  50 

3.4487 

3.4489 

3.4490 

3.4492 

3.4493 

3.4495 

3.4496 

3.4498 

3.4499 

3.4501 

0  47   0 

3.4502 

3.4504 

3.4506 

3.4507 

3.4509 

3.4510 

3.4512 

3.4513 

3.4515 

3.4516 

47  10 

3.4518 

3.4519 

3.4521 

3.4522 

3.4524 

3.4526 

3.4527 

3.4529 

3.4530 

3.4532 

47  20 

3.45331  3.4535 

3.4536 

3.4538 

3.45-39 

3.4541 

3.4542 

3.4544 

3.4545 

3.4547 

47  30 

3.45481  3.4550 

3.4551 

3.4553 

3.4555 

3.4556 

3.4558 

3.4559  !  3.4561 

3.4562 

47  40 

3.4564  3.4565 

3.4567 

3.4568 

3.4570 

3.  4  5  VI 

3.4573 

3.4574  3.4576 

3.4577 

v47  50 

3.4579 

3.4580 

3.4582 

3.4583 

3.4585 

3.4586 

3.4588 

3.4589  3.4591 

3.4592 

0  48   0 

3.4594 

3.4595 

3.4597 

3.4598;  3.4600 

3.4601 

3.4603 

3.4604  3.4606 

3.4607 

48  10 

3.4609  3.4610 

3.4612 

3.4613 

3.4615 

3.4616 

3.4618 

3.4619  3.4621 

3.4622 

48  20 

3.4624J  3.4625 

3.4627 

3.4628 

3.4630 

3.4631 

3.4633 

3.4634  3.4636 

3.4637 

48  30 

3.46391  3.4640 

3.4642 

3.4643 

3.4645 

3.4646 

3.4648 

3.4649!  3.4651 

3.4652 

48  40 

3.4654  i  3.4655 

3.4657 

3.4658 

3.4660 

3.4661 

3.4663 

3.4664'  3.4666 

3.4667 

48  50 

3.4669 

3.4670 

3.4672 

3.4673 

3.4675 

3.4676 

3.4678 

3.4679,  3.4681 

3.4682 

0  49   0 

3.4683 

3.4685 

3.4686 

.3.4688 

3.4689 

3.4691 

3.4692 

3.4694  3.4695 

3.4697 

49  10 

3.4698  3.4700 

3.4701 

3.4703 

3.4704 

3.4706 

3.4707 

3.4709  3.4710  3.4711 

49  20 

3.4713!  3.4714 

3.4716 

3.4717 

3.4719 

3.4720!  3.4722 

3.4723  3.4725  3.4726 

49  30 

3.4728,  3.4729 

3.4730 

3.4732 

34733 

3.4735!  3.4736 

3.4738  3.4739  1  3.4741 

49  40 

3.4742  3.4744 

3.4745 

3.4747 

3.4748 

3.47491  3.4751 

3.4752  3.4754|  3.4755 

49  50 

3.47571  3.4758 

3.4760 

3.4781 

3.4763 

34764  3.4765 

3.4767  3.4768!  3.4770 

44: 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

d 

it 

it 

2 

it 

3 

4 

ii 

5 

6 

u 

7 

u 

8 

9 

Oh'50m-  0s- 

3.4771 

3.4773 

3.4774 

3.4776 

3.4777 

3.4778 

3.4780 

3.4781 

3.4783 

3.4784 

50  10 

3.4786 

3.4787 

3.4789 

3.4790 

3.4791 

3.4793 

3.4794 

3.4796 

3.4797 

3.4799 

50  20 

3.4800 

3.4802 

3.4803 

3.4804 

3.4806 

3.4807 

3.4809 

3.4810 

3.4812 

3.4813 

50  30 

3.4814 

3.4816 

3.4817 

3.4819 

3.4820 

3.4822 

3.4823 

3.4824 

3.4826 

3.4827 

50  40 

3.4829 

3.4830 

3.4832 

3.4833 

3.4834 

3.4836 

3.4837 

3.4839 

3.4840 

3.4842 

50  50 

3.4843 

3.4844 

3.4846 

3.4847 

3.4849 

3.4850 

3.4852 

3.4853 

3.4854 

3.4856 

0  51   0 

3.4857 

3.4859 

3.4860 

3.4861 

3.4863 

3.4864 

3.4866 

3.4867 

3.4869 

3.4870 

51  10 

3.4871 

3.4873 

3.4874 

3.4876 

3.4877 

3.4878 

3.4880 

3.4881 

3.4883 

3.4884 

51  20 

3.4886 

3.4887 

3.4888 

3.4890 

3.4891 

3.4893 

3.4894 

3.4895 

3.4897 

3.4898 

51  30 

.3.4900 

3.4901 

3.4902 

3.4904 

3.4905 

3.4907 

3.4908 

3.4909 

3.4911 

3.4912 

51  40 

3.4914 

3.4915 

3.4916 

3.4918 

3.4919 

3.4921 

3.4922 

3.4923 

3.4925 

3.4926 

51  50 

3.4928 

3.4929 

3.4930 

3.4932 

3.4933 

3.4935 

3.4936 

3.4937 

3.4939 

3.4940 

0  52   0 

34942 

3.4943 

3.4944 

3.4946 

3.4947 

3.4949 

3.4950 

3.4951 

3.4953 

3.4954 

52  10 

3.4955 

3.4957 

3.4958 

3.4960 

3.49  01 

3.4962 

3.4964 

3.4965 

3.4967 

3.4968 

52  20 

3.4969 

3.4971 

3.4972 

3.4973 

3.4975 

3.4976 

3.4978 

3.4979 

3.4980 

3.4982 

52  30 

3.4983 

3.4984 

3.4986 

3.4987 

3.4989 

3.4990 

3.4991 

3.4993 

3.4994 

3.4995 

52  40 

3.4997 

3.4998 

3.5000 

3.5001 

3.5002 

3.5004 

3.5005 

3.5006 

3.5008 

3.5009 

52  50 

3.5011 

35012 

3.5013 

3.5015 

3.5016 

3.5017 

3.5019 

3.5020 

3.5022 

3.5023 

0  53   0 

3.5024 

3.5026 

3.5027 

3.5028 

3.5030 

3.5031 

3.5032 

3.5034 

3.5035 

3.5037 

53  10 

3.5038 

3.5039 

3.5041 

3.5042 

3.5043 

3.5045 

3.5046 

3.5047 

3.5049 

3.5050 

53  20 

3.5051 

3.5053 

3.5054 

3.5056 

3.5057 

3.5058 

3.5060 

3.5061 

3.5062 

3.5064 

53  30 

3.5065 

3.5066 

3.5068 

3.5069 

3.5070 

3.5072 

3.5073 

3.5075 

3.5076 

3.5077 

53  40 

3.5079 

3.5080 

3.5081 

3.5083 

3.5084 

3.5085 

3.5087 

3.5088 

3.5089 

3.5091 

53  50 

3.5092 

3.5093 

3.5095 

3.5096 

3.5097 

3.5099 

3.5100 

3.5101 

3.5103 

35104 

0  54   0 

3.5105 

3.5107 

3.5108 

3.5109 

3.5111 

3.5112 

3.5113 

3.5115 

3.5116 

3.5117 

54  10 

3.5119 

3.5120 

3.5122 

3.5123 

3.5124 

3.5126 

3.5127 

3.5128 

3.5130 

3.5131 

54  20 

3.5132 

3.5134 

3.5135 

3.5136 

3.5138 

3.5139 

3.5140 

3.5141 

3.51431  3.5144 

54  30 

3.5145 

3.5147 

3.5148 

3.5149 

3.5151 

3.5152 

3.5153 

3.5155 

3.5156 

3.5157 

54  40 

3.5159 

3.5160 

3.5161 

3.5163 

3.5164 

3.5165 

3.5167 

3.5168 

3.5169 

3.5171 

54  50 

3.5172 

3.5173 

3.5175 

3.5176 

3.5177 

3.5179 

3.5180 

3.5181 

3.5183 

3.5184 

0  55   0 

3.5185 

3.5186 

3.5188 

3.5189 

3.5190 

3.5192 

3.5193 

3.5194 

3.5196 

3.5197 

55  10 

3.5198 

3.5200 

3.5201 

3.5202 

3.5204 

3.5205 

3.5206 

3.5207 

3.5209 

3.5210 

55  20 

3.5211 

3.5213 

3.5214 

3.5215 

3.5217 

3.5218 

3.5219 

3.5221 

3.5222 

3.5223 

55  30 

3.5224 

3.5226 

3.5227 

3.5228 

3.5230 

8.5231 

3.5232 

3.5234 

3.5235 

3.5236 

55  40 

3.5237 

3.5239 

3.5240 

3.5241 

3.5243 

3.5244 

3.5245 

3.5247 

3.5248 

3.5249 

55  50 

3.5250 

3.5252 

3.5253 

3.5254 

3.5256 

3.5257 

3.5258 

3.5260 

3.5261 

3.5262 

0  56   0 

3.5263 

3.5265 

3.5266 

3.5267 

3.5269 

3.5270 

3.5271 

3.5272 

3.5274 

3.5275 

56  10 

3.5276 

3.5278 

3.5279 

3.5280 

3.5281 

3.5283 

3.5284 

3.5285 

3.5287 

3.5288 

56  20 

3.5289 

3.5290 

3.5292 

3.5293 

3.5294 

3.5296 

3.5297 

3.5298 

3.5299 

3.5301 

56  30 

3.5302 

3.5303  3.5305 

3.5306 

3.5307 

3.5308 

3.5310 

3.5311 

3.5312 

3.5314 

56  40 

3.5315 

3.5316  3.5317 

3.5319 

3.5320 

3.5321 

3.5322 

3.5324 

3.5325 

3.5326 

56  50 

3.5328 

3.5329  3.5330 

3.5331 

3.5333 

3.5334 

3.5335 

3.5336 

3.5338 

3.5339 

0  57   0 

3.5340 

3.5342  3.5343 

3.5344 

3.5345 

3.5347 

3.5348 

3.5349 

3.5350 

3.5352 

57  10 

3.5353 

3.5354  3.5355 

3.5357 

3.5358 

3.5359 

3.5361 

3.5362 

3.5363 

3.5364 

57  20 

3.5366 

3.5367  3.5368 

3.5369 

3.5371 

3.5372 

3.5373 

3.5374 

3.5376 

3.5377 

57  30 

3.5378 

3.5379 

3.5381 

3.5382 

3.5383 

3.5384 

3.5386 

3.5387 

3.5388 

3.5390 

57  40 

3.5391 

3.5392 

3.5393 

3.5395 

3.5396 

3.5397 

3.5398 

3.5400 

3.5401 

3.5402 

57  50 

3.5403 

3.5405 

3.5406 

3.5407 

3.5408 

3.5410 

3.5411 

3.5412 

3.5413 

3.5415 

0  58   0 

3.5416 

3.5417 

3.5418 

3.5420 

3.5421 

3.5422 

3.5423 

3.5425 

3.5426 

3.5427 

58  10 

3.5428 

3.5429  1  3.5431 

3.5432 

3.5433 

3.5434 

35436 

3.5437 

3.5438!  3.5439 

58  20 

3.5441 

3.5442!  3.5443 

3.5444 

3.5446 

3.5447 

3.5448 

3.5449 

3.5451  3.5452 

58  30 

3.5453 

3.5454 

3.5456 

3.5457 

3.5458 

3.5459 

3.5460 

3.5462 

3.5463;  3.5464 

58  40 

3.5465 

3.5467 

3.5468 

3.5469 

3.5470 

3.5472 

3.5473 

3.5474 

3.5475 

3.5477 

58  50 

3.5478 

3.5479 

3.5480 

3.5481 

3.5483 

3.5484 

3.5485 

35486 

3.5488 

3.5489 

0  59   0 

3.5490 

3.5491 

3.5492 

3.5494 

3.5495 

3.5496 

3.5497 

3.5499 

3.5500 

3.5501 

59  10 

3.5502 

3.5504 

3.5505 

3.5506 

3.5507 

3.5508 

3.5510 

3.5511 

3.5512 

3.5513 

59  20 

3.5514 

3.5516 

3.5517 

3.5518 

3.5519 

3.5521 

3.5522 

3.5523 

3.5524 

3.5525 

59  30 

3.5527 

3.5528 

3.5529 

3.5530 

3.5532 

3.5533 

3.5534 

3.5535 

3.5536 

3.5538 

59  40 

3.5539 

3.5540 

3.5541 

3.5542 

3.5544 

3.5545 

3.5546 

3.5547 

3.5549 

3.5550 

59  50 

3.5551 

3.5552 

3.5553 

3.5555 

3.5556 

3.5557 

3.5558 

3.5559 

3.5561 

3.5562 

45 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

E 

0 

i 

2 

n 

3 

4 

5 

6 

7 

8 

n 

9 

I  °,  i  n 

ij  1*  O'n>  O3- 

3.5563 

3.5564 

3.5565 

3.5567 

3.5568 

3.5569 

3.5570 

3.5571 

3.5573 

3.5574 

0  10 

3.5575 

3.5576 

3.5577 

3.5579 

3.5580 

3.5581 

3.5582 

3.5583 

3.5585 

3.5586 

0  20 

3.5587 

3.5588 

3.5589 

3.5591 

3.5592 

3.5593 

3.5594 

3.5595  3.5597 

3.5598 

0  30 

3.5599 

3.5600 

3.5601 

3.5603 

3.5604 

3.5605 

3.5606 

3.5607 

3.5609 

3.5610 

0  40 

3.5611 

3.5612 

3.5613 

3.5615 

3.5616 

3.5617 

3.5618 

3.5619 

3.5621 

3.5622 

0  50 

3.5623 

3.5624 

3.5625 

3.5626 

3.5628 

3.5629 

3.5630 

3.5631 

3.5632 

3.5634 

1     0 

3.5635 

3.5636 

3.5637 

3.5638 

3.5640 

3.5641 

3.5642 

3.5643 

3.5644 

35645 

10 

3.5647 

3.5648 

3.5649 

3.5650 

3.5651 

3.5653 

3.5654 

3.5655 

3.5656 

3.5657 

20 

3.5658 

3.5660 

3.5661 

3.5662 

3.5663 

3.5664 

3.5666 

3.5667 

3.5668 

3.5669 

30 

3.5670 

3.5671 

3.5673 

3.5674 

3.5675 

3.5676 

3.5677 

3.5678 

3.5680 

3.5681 

40 

3.5682 

3.5683 

3.5684 

3.5686 

3.5687 

3.5688 

3.5689 

3.5690 

3.5691 

3.5693 

50 

3.5694 

3.5695 

3.5696 

35697 

3.5698 

3.5700 

3.5701 

3.5702 

3.5703 

3.5704 

1   2   0 

3.5705 

3.5707 

3.5708 

3.5709 

3.5710 

3.5711 

3.5712 

3.5714 

3.5715 

3.5716 

2  10 

3.5717 

3.5718 

3.5719 

3.5721 

3.5722 

3.5723 

3.5724 

3.5725 

3.5726 

35728 

2  20 

3.5729 

3.5730 

3.5731 

3.5732 

3.5733 

3.5735 

3.5736 

3.5737 

3.5738 

3.5739 

2  30 

3.5740 

3.5741 

3.5742 

3.5744 

3.5745 

3.5746 

3.5747 

3.5748 

3.5750 

3.5751 

2  40 

3.5752 

3.5753 

3.5754 

3.5755 

3.5756 

3.5758 

3.5759 

3.5760 

3.5761 

3.5762 

2  50 

3.5763 

3.5765 

3.5766 

3.5767 

3.5768 

3.5769 

3.5770 

3.5771 

3.5773 

3.5774 

1   3   0 

3.5775 

3.5776 

3.5777 

3.5778 

3.5780 

3.5781 

3.5782 

3  5783 

3.5784 

3.5785 

3  10 

3.5786 

3.5788 

3.5789 

3.5790 

3.5791 

3.5792 

3.5793 

3.5794 

3.5796 

3.5797 

3  20 

3.5798 

3.5799 

3.5800 

3.5801 

3.5802 

3.5804 

3.5805 

3.5806 

3.5807 

3.5808 

3  30 

3.5809 

3.5810 

3.5812 

3.5813 

3.5814 

3.5815 

3.5816 

3.5817 

3.5818 

3.5819 

3  40 

3.5821 

3.5822 

3.5823 

3.5824 

3.5825 

3.5826 

3.5827 

3.5829 

3.5830 

3.5831 

3  50 

3.5832 

3.5833 

3.5834 

3.5835 

3.5837 

3.5838 

3.5839 

3.5840 

3.5841 

3.5842 

1   4   0 

3.5843 

3.5844 

3.5846 

3.5847 

3.5848 

3.5849 

3.5850 

3.5851 

3.5852 

3.5853 

4  10 

3.5855 

3.5856 

3.5857 

3.5858 

3.5859 

3.5860 

3.5861 

3.5862 

3.5864 

3.5865 

4  20 

3.5866 

3.5867 

3.5868 

3.5869 

3.5870 

3.5871 

3.5873 

3.5874 

3.5875 

3.5876 

4  30 

3.5877 

3.5878 

3.5879 

3.5880 

3.5882 

3.5883 

3.5884 

3.5885 

3.5886 

3.5887 

4  40 

3.5888 

3.5889 

3.5891 

3.5892 

3.5893 

3.5894 

3.5895 

3.5896 

3.5897 

3.5898 

4  50 

3.5899 

3.5901 

3.5902 

3.5903 

3.5904 

3.5905 

3.5906 

3.5907 

3.5908 

3.5910 

1   5   0 

3.5911 

3.5912 

3.5913 

3.5914 

3.5915 

3.5916 

3.5917 

3.5918 

3.5920 

3.5921 

5  10 

3.5922 

3.5923 

3.5924 

3.5925 

3.5926 

3.5927 

3.5928 

3.5930 

3.5931 

3.5932 

5  20 

3.5933 

3.5934 

3.5935 

3.5936 

3.5937 

3.5938 

3.5940 

3.5941 

3.5942 

3.5943 

5  30 

3.5944 

3.5945 

3.5946 

3.5947 

3.5948 

3.5949 

3.5951 

3.5952 

3.5953 

3.5954 

5  40 

3.5955 

3.5956 

3.5957 

3.5958 

3.5959 

3.5960 

3.5962 

3.5963 

3.5964 

3.5965 

5  50 

3.5966 

3.5967 

3.5968 

3.5969 

3.5970 

3.5971 

3.5973 

3.5974 

3.5975 

3.5976 

1   6   0 

3.5977 

3.5978 

3.5979 

3.5980 

3.5981 

3.5982 

3.5984 

3.5985 

3.5986 

3.5987 

6  10 

3.5988 

3.5989 

3.5990 

3.5991 

3.5992 

3.5993 

3.5994 

3.5996 

3.5997 

3.5998 

6  20 

3.5999 

3.6000 

3.6001 

3.6002 

3.6003 

3.6004 

3.6005 

3.6006 

3.6008 

3.6009 

6  30 

3.6010 

3.6011 

3.6012 

3.6013 

3.6014 

3.6015 

3.6016 

3.6017 

3.60i8 

3.6020 

6  40 

3.6021 

3.6022 

3.6023 

3.6024 

3.6025 

3.6026 

3.6027 

3.6028 

3.6029 

3.6030 

6  50 

3.6031 

3.6033 

3.6034 

3.6035 

3.6036 

3.6037 

3.6038 

3.6039 

3.6040 

3.6041 

1   7   0 

3.6042 

3.6043 

3.6044 

3.6046 

3.6047 

3.6048 

3.6049 

3.6050 

3.6051 

3.6052 

7  10 

3.6053 

3.6054 

3.6055 

3.6056 

3.6057 

3.6058 

3.6060 

3.6061 

3.6062 

3.6063 

7  20 

3.6064 

3.6065 

3.6066 

3.6067 

3.6068 

3.6069 

3.6070 

3.6071 

3.6072 

3.6073 

7  30 

3.6075 

3.6076 

3.6077 

3.6078 

3.6079 

3.6080 

3.6081 

3.6082 

3.6083 

3.6084 

7  40 

3.6085 

3.6086 

3.6087 

3.6088 

3.6090 

3.6091 

3.6092 

3.6093 

3.6094 

3.6095 

7  50 

3.6096 

3.6097 

3.6098 

3.6099 

3.6100 

3.6101 

3.6102 

3.6103 

3.6104 

3.6106 

1   8   0 

3.6107 

3.6108 

3.6109 

3.6110 

3.6111 

3.6112 

3.6113 

3.6114 

3.6115 

3.6116 

8  10 

3.6117 

3.6118 

3.6119 

3.6120 

3.6121 

3.6123 

3.6124 

3.6125 

3.6126 

3.6127 

8  20 

3.6128 

3.6129 

3.6130 

3.6131 

3.6132 

3.6133 

3.6134 

3.6135 

3.6136 

3.6137 

8  30 

3.6138 

3.6139 

3.6141 

3.6142 

3.6143 

3.6144 

3.6145 

3.6146 

3.6147 

3.6148 

8  40 

3.6149 

3.6150 

3.6151 

3.6152 

3.6153 

3.6154 

3.6155 

3.6156 

3.6157 

3.6158 

8  50 

3.6160 

3.6161 

3.6162 

3.6163 

3.6164 

3.6165 

3.6166 

3.6167 

3.6168 

3.6169 

1   9   0 

3.6170 

3.6171 

3.6172 

3.6173 

3.6174 

3.6175 

3.6176 

3.6177 

3.6178 

3.6179 

9  10 

3.6180 

3.6182 

3.6183 

3.6184 

3.6185 

3.6186 

3.6187 

3.6188 

3.6189 

3.6190 

9  20 

3.6191 

3.6192 

3.6193 

3.61941  3.6195 

3.6196 

3.6197 

3.6198 

3.6199 

3.6200 

9  30 

3.6201 

3.6202 

3.6203 

3.6204'  3.6206 

3.6207 

3.6208 

3.6209 

3.6210 

3.6211 

9  40 

3.6212 

3.6213 

3.6214 

3.6215  3.6216 

3.6217 

3.6218 

3.6219 

3.6220 

3.6221 

9  50 

3.6222 

3.6223 

3.6224 

3.6225  '  3.6226 

3.6227 

3.6228 

3.6229 

3.6230 

3.6231 

TABLE    IX. 


(LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

£ 

0 

i 

& 

3 

// 

4 

5 

/( 

6 

7 

// 

8 

9 

°l_    1    II 

ih-iom-  os- 

3.6232 

3.6234 

3.6235 

3.6236 

3.6237 

3.6238 

3.6239 

3.6240 

3.6241 

3.6242 

10  10 

3.6243 

3.6244 

3.6245 

3.6246 

3.6247 

3.6248 

3.6249 

3.6250 

3.6251 

3.6252 

10  20 

3.6253 

3.6254 

3.6255 

3.6256 

3.6257 

3.6258 

3.6259 

3.6260 

3.6261 

3.6262 

10  30 

3.6263 

3.6264 

3.6265 

3.6266 

3.6268 

3.6269 

3.6270 

3.6271 

3.6272 

3.6273 

10  40 

3.6274 

3.6275 

3.6276 

3.6277 

3.6278 

3.6279 

3.6280 

3.628] 

3.6282 

3.6283 

10  50 

3.6284 

3.6285 

3.6286 

3.6287 

3.6288 

3.6289 

3.6290 

3.6291 

3.6292 

3.6293 

1  11   0 

3.6294 

3.6295 

3.6296 

3.6297 

3.6298 

3.6299 

3.6300 

3.6301 

3.6302 

3.6303 

11  10 

3.6304 

3.6305 

3.6306 

3.6307 

3.6308 

3.6309 

3.6310 

3.6311 

36312 

3.6313 

11  20 

3.6314 

3.6315 

3.6316 

3.6317 

3.6318 

3.6320 

3.6321 

3.6322 

3.6323 

3.6324 

11  30 

3.6325 

3.6326 

3.6327 

3.6328 

3.6329 

3.6330 

3.6331 

3.6332 

3.6333 

3.6334 

11  40 

3.6335 

3.6336 

3.6337 

3.6338 

3.6339 

3.6340 

3.6341 

3.6342 

3.6343 

3.6344 

11  50 

3.6345 

3.6346 

3.6347 

3.6348 

3.6349 

3.6350 

3.6351 

3.6352 

3.6353 

3.6354 

1  12   0 

3.6355 

3.6356 

3.6357 

3.6358 

3.6359 

3.6360 

3.6361 

3.6362 

3.6363 

3.6364 

12  10 

3.6365 

3.6366 

3.6367 

3.6368 

3.6369 

3.6370 

3.6371 

3.6372 

3.6373 

3.6374 

12  20 

3.6375 

3.6376 

3.6377 

3.6378 

3.6379 

3.6380 

3.6381 

3.6382 

3.6383 

3.6384 

12  30 

3.6385 

3.6386 

3.6387 

3.6388 

3.6389 

3.6390 

3.6391 

3.6392 

3.6393 

3.6394 

12  40 

3.6395 

3.6396 

3.6397 

3.6398 

3.6399 

3.6400 

3.6401 

3.6402 

3.6403 

3.6404 

12  50 

3.6405 

3.6406 

3.6407 

3.6408 

3.6409 

3.6410 

3.6411 

3.6412 

3.6413 

3.6414 

1  1,3   0 

3.6415 

3.6416 

3.6417 

3.6418 

3.6419 

3.6420 

3.6421 

3.6422 

3.6423 

3.6424 

13  10 

3.6425 

3.6426 

3.6427 

3.6428 

3.6429 

3.6430 

3.6431 

3.6432 

3.6433 

3.6434 

13  20 

3.6435 

3.6436 

3.6437 

3.6437 

3.6438 

3.6439 

3.6440 

3.6441 

3.6442 

3.6443 

13  30 

3.6444 

3.6445 

3.6446 

3.6447 

3.6448 

3.6449 

3.6450 

3.6451 

3.6452 

3.6453 

13  40 

3.6454 

3.6455 

3.6456 

3.6457 

3.6458 

3.6459 

3.6460 

3.6461 

3.6462 

3.6463 

13  50 

3.6464 

3.6465 

3.6466 

3.6467 

3.6468 

3.6469 

3.6470 

3.6471 

3.6472 

3.6473 

1  14   0 

3.6474 

3.6475 

3.6476 

3.6477 

3.6478 

3.6479 

3.6480 

3.6481 

3.6482 

3.6483 

14  10 

3.6484 

3.6485 

3.6486 

3.6487 

3.6488 

3.6488 

3.6489 

3.6490 

3.6491 

3.6492 

14  20 

3.6493 

3.6494 

3.6495 

3.6496 

3.6497 

3.6498 

3.6499 

3.6500 

3.6501 

3.6502 

14  30 

3.6503 

3.6504 

3.6505 

3.6506 

3.6507 

3.6508 

3.6509 

3.6510 

3.6511 

3.6512 

14  40 

3.6513 

3.6514 

3.6515 

3.6516 

3.6517 

3.6518 

3.6519 

3.6520 

3.6521 

3.6521 

14  50 

3.6522 

3.6523 

3.6524 

3.6525 

3.6526 

3.6527 

3.6528 

3.6529 

3.6530 

3.6531 

1  15   0 

3.6532 

3.6533 

3.6534 

3.6535 

3.6536 

3.6537 

3.6538 

3.6539 

3.6540 

3.6541 

15  10 

3.6542 

3.6543 

3.6544 

3.6545 

3.6546 

3.6547 

3.6548 

3.6549 

3.6549 

3.6550 

15  20 

3.6551 

3.6552 

3.6553 

3.6554 

3.6555 

3.6556 

3.6557 

3.6558 

3.6559 

3.6560 

15  30 

3.6561 

3.6562 

3.6563 

3.6564 

3.6565 

3.6566 

3.6567 

3.6568 

3.6569 

3.6570 

15  40 

3.6571 

3.6572 

3.6572 

3.6573 

3.6574 

3.6575 

3.6576 

3.6577 

3.6578 

3.6579 

15  50 

3.6580 

3.6581 

3.6582 

3.6583 

3.6584 

3.6585 

3.6536 

3.6587 

3.6588 

3.6589 

1  16   0 

3.6590 

3.6591 

3.6592 

3.6593 

3.6593 

3.6594 

3.6595 

3.6596 

3.6597 

3.6598 

16  10 

3.6599 

3.6600 

3.6601 

3.6602 

3.6603 

3.6604 

3.6605 

3.6606 

3.6607 

3.6608 

16  20 

3.6609 

3.6610 

3.6611 

3.6611 

3.6612 

3.6613 

3.6614 

3.6615 

3.6616 

3.6617 

16  30 

3.6618 

3.6619 

3.6620 

3.6621 

3.6622 

3.6623 

3.6624 

3.6625 

3.6626 

3.6627 

16  40 

3.6628 

3.6629 

3.6629 

3.6630 

3.6631 

36632 

3.6633 

3.6634 

3.6635 

3.6636 

16  50 

3.6637 

3.6638 

3.6639 

3.6640 

3.6641 

3.6642 

3.6643 

3.6644 

3.6645 

3.6645 

1  17   0 

3.6646 

3.6647 

3.6648 

3.6649 

3.6650 

3.6651 

3.6652 

3.6653 

3.6654 

3.6655 

17  10 

3.6656 

3.6657 

3.6658 

3.6659 

3.6660 

3.6660 

3.6661 

3.6662 

3.6663 

3.6664 

17  20 

3.6665 

3.6666 

3.6667 

3.6668 

3.6669 

3.6670 

3.6671 

3.6672 

3.6673 

3.6674 

17  30 

3.6675 

3.6675 

3.6676 

3.6677 

3.6678 

3.6679 

3.6680 

3.6681 

3.6682 

3.6683 

17  40 

:  3.6684 

3.6685 

3.6686 

3.6687 

3.6688 

3.6689 

3.6689 

3.6690 

3.6691 

3.6692 

17  50 

3.6693 

3.6694 

3.6695 

3.6696 

3.6697 

3.6698 

3.6699 

3.6700 

3.6701 

3.6702 

1  18   0 

3.6702 

3.6703 

3.6704 

3.6705 

3.6706 

3.6707 

3.6708 

3.6709 

3.6710 

3.6711 

18  10 

3.6712 

3.6713 

3.6714 

3.6715 

3.6715 

3.6716 

3.6717 

3.6718 

3.6719 

3.6720 

18  20 

3.6721 

3.6722 

3.6723 

3.6724 

3.6725 

3.6726 

3.6727 

3.6727 

3.6728 

3.6729 

18  30 

3.6730 

3.6731 

3.6732 

3.6733 

3.6734 

3.6735 

3.6736 

3.6737 

3.6738 

3.6738 

18  40 

3.6739 

3.6740 

3.6741 

3.6742 

36743 

3.6744 

3.6745 

3.6746 

3.6747 

3.6748 

18  50 

3.6749 

3.6750 

3.6750 

3.6751 

3.o752 

3.6753 

3.6754 

3.6755 

3.6756 

3.6757 

1  19   0 

3.6758 

3.6759 

3.6760 

3.6761 

3.6761 

3.6762 

3.6763 

3.6764 

3.6765 

3.6766 

19  10 

3.6767 

3.6768 

3.6769 

3.6770 

3.6771 

3.6772 

3.6772 

3.6773 

3.6774 

3.6775 

19  20 

3.6776 

3.6777 

3.6778 

3.6779 

3.6780 

3.6781 

3.6782 

3.6782 

3.6783 

3.6784 

19  30 

3.6785 

3.6786 

3.6787 

3.6788 

3.6789 

3.6790 

3.6791 

3.6792 

3.6792 

3.6793 

19  40 

3.6794 

3.6795 

36796 

3.6797 

3.6798 

3.6799 

3.6800 

3.6801 

3.6802 

3.6802 

19  50 

3.6803 

3.6804 

3.6805 

3.6806 

3.6807 

3.6808  3.6809 

3.6810 

3.6811 

3.6812 

4:7 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

0 

i 

\ 

3 

// 

4 

1 

5 

ii 

6 

7 

8 

9 

O    i     ,, 

lh.20m.  QS. 

3.6812 

3.6813 

3.6814  3.6815 

3.6816 

3.6817 

3.6818 

3.6819 

3.6820 

36821 

20  10 

3.6821 

3.6822 

3.6823  3.6824 

3.6825 

3.6826 

3.6827 

3.6828 

3.6829 

3.6830 

20  20 

3.6830 

3.6831 

3.6832 

3.6833 

3.6834 

3.6835 

3.6836 

3.6837 

3.6838 

3.6839 

20  30 

3.6839 

3.6840 

3.6841 

3.6842 

3.6843 

3.6844 

3.6845 

3.6846 

3.6847 

3.6848 

20  40 

3.6848 

3.6849 

3.6850 

3.6851 

3.6852 

3.6853 

3.6854 

3.6855 

3.6856 

3.6857 

20  50 

3.6857 

3.6858 

3.6859 

3.6860 

3.6861 

3.6862 

3.6863 

3.6864 

3.6865 

3.6865 

1  21   0 

3.6866 

3.6867 

3.6868 

3.6869 

3.6870 

3.6871 

3.6872 

3.6873 

3.6874 

3.6874 

21  10 

3.6875 

3.6876 

3.6877 

3.6878 

3.6879 

3.6880 

3.6881 

3.6882 

3.6882 

3.6883 

21  20 

3.6884 

3.6885 

P.6886 

3.6887 

3.6888 

3.6889 

3.6890 

3.6890 

3.6891 

3.6892 

21  30 

3.6893 

3.6894 

3.6895 

3.6896 

3.6897 

3.6898 

3.6898 

3.6899 

3.6900 

3.6901 

21  40 

3.6902 

3.6903 

3.6904 

3.6905 

3.6906 

3.6906 

3.6907 

3.6908 

3.6909 

3.6910 

21  50 

3.6911 

3.6912 

3.6913 

3.6913 

3.6914 

3.6915 

3.6916 

3.6917 

3.6918 

3.6919 

1  22   0 

3.6920 

3.6921 

3.6921 

3.6922 

3.6923 

3.6924 

3.6925 

3.6926 

3.6927 

3.6928 

22  10 

3.6928 

3.6929 

3.6930 

3.6931 

3.6932 

3.6933 

3.6934 

3.6935 

3.6936 

3.6936 

22  20 

3.6937 

3.6938 

3.6939 

3.6940 

3.6941 

3.6942 

3.6943 

3.6943 

3.6944 

3.6945 

22  30 

3.6946 

3.3947 

3.6948 

3.6949 

3.6950 

3.6950 

3.6951 

3.6952 

3.6953 

3.6954 

22  40 

3.6955 

3.6956 

3.6957 

3.6957 

3.6958 

3.6959 

.3.6960 

3.6961 

3.6962 

3.6963 

22  50 

3.6964 

3.6964 

3.6965 

3.6966 

3.6967 

3.6968 

3.6969 

3.6970 

3.6971 

3.6971 

1  23   0 

3.6972 

3.6973 

3.6974 

3.6975 

3.6976 

3.6977 

3.6978 

3.6978 

3.6979 

3.6980 

23  10 

3.6981 

3.6982 

3.6983 

3.6984 

3.6984 

3.6985 

3.6986 

3.6987 

3.6988 

3.6989 

23  20 

3.6990 

3.6991 

3.6991 

3.6992 

3.6993 

3.6994 

3.6995 

3.6996 

3.6997 

3.6998 

23  30 

3.6998 

3.6999 

3.7000 

3.7001 

3.7002 

3.7003 

3.7004 

3.7004 

3.7005 

3.7006 

23  40 

3.7007 

3.7008 

3.7009 

3.7010 

3.7010 

3.7011 

3.7012 

3.7013 

3.7014 

3.7015 

23  50 

3.7016 

3.7017 

3.7017 

3.7018 

3.7019 

3.7020 

3.7021 

3.7022 

3.7023 

3.7023 

1  24   0 

3.7024 

3.7025 

3.7026  3.7027 

3.7028 

3.7029 

3.7029 

3.7020 

3.7031 

3.7032 

24  10 

3.7033 

3.7034 

3.7035  3.7035 

37036 

3.7037 

3.7038 

3.7039 

3.7040  3.7041 

24  20 

3.7042 

3.7042 

3.7043  3.7044 

3.7045 

3.7046 

3.7047 

3.7048 

3.7048  !  3.7049 

24  30 

3.7050 

3.7051 

3.7052 

3.7053 

3.7054 

3.7054 

3.7055 

3.7056 

3.7057 

3.7058 

24  40 

3.7059 

3.7060 

3.7060 

3.7061 

3.7062 

3.7063 

3.7064 

3.7065 

3.7065 

3.7066 

24  50 

3.7067 

3.7068 

3.7069 

3.7070 

3.7071 

37071 

3.7072 

3.7073 

3.7074 

3.7075 

1  25   0 

3.7076 

3.7077 

3.7077 

3.7078 

3.7079 

3.7080 

3.7081 

3.7082 

3.7083 

3.70S3 

25  10 

3.7084 

3.7085 

3.7086 

3.7087 

3.7088 

3.7088 

3.7089 

3.7090 

3.7091 

3.7092 

25  20 

3.7093 

3.7094 

3.7094 

3.7095 

3.7096 

3.7097 

3.7098  3.7099 

3.7099  3.7100 

25  30 

3.7101 

3.7102 

3.7103 

3.7104 

3.7105 

3.7105 

3.7106  3.7107 

3.7108!  3.7109 

25  40 

3.7110 

3.7110 

3.7111 

3.7112 

3.7113 

3.7114 

3.7115 

3.7116 

3.7116!  3.7117 

25  50 

3.7118 

3.7119 

3.7120 

3.7121 

3.7121 

3.7122 

3.7123 

3.7124 

3.7125  3.7126 

1  26   0 

3.7126 

3.7127 

3.7128 

3.7129 

3.7130 

3.7131 

3.7132 

3.7132 

3.7133!  3.7134 

26  10 

3.7135 

3.7136 

3.7137 

3.7137 

3.7138 

3.7139 

3.7140 

3.7141 

3.7142  3.7142 

26  20 

3.7143 

3.7144 

3.7145 

3.7146 

3.7147 

3.7147 

3.7148 

3.7149 

3.7150  3.7151 

26  30 

3.7152 

3.7153  3.7153 

3.7154 

3.7155 

3.7156 

3.7157 

3.7158 

3.7159!  3.7159 

26  40 

3.7160 

3.7161  3.7162 

3.7163 

3.7163 

3.7164 

3.7165 

3.7166 

3.7167 

3.7168 

26  50 

3.7168 

3.7169  3.7170 

3.7171 

3.7172 

3.7173 

3.7173 

3.7174 

3.7175 

3.7176 

1  27   0 

3.7177 

3.7178  3.7178 

3.7179 

3.7180 

3.7181 

3.7182 

3.7183 

3.7183 

3.7184 

27  10 

3.7185 

3.7186  3.7187 

3.7188 

3.7188 

3.7189 

3.7190 

3.7191 

3.7192  3.7192 

27  20 

3.7193 

3.7194  3.7195 

3.7196 

3.7197 

3.7197 

37198 

3.7199 

3.7200  3.7201 

27  30 

3.7202 

3.7202  3.7203 

3.7204 

3.7205 

3.7206 

3.7207 

3.7207 

3.7208  3.7209 

27  40 

3.7210 

3.7211  3.7212 

3.7212 

3.7213 

3.7214 

3.7215 

3.7216 

3.7216  3.7217 

27  50 

3.7218 

3.7219 

3.7220 

3.7221 

3.7221 

3.7222 

3.7223 

3.7224 

3.7225  3.7226 

1  28   0 

3.7226 

3.7227 

3.7228 

3.7229 

3.7230 

3.7230 

3.7231 

3.7232 

3.7233!  3.7234 

28  10 

3.7235 

3.7235 

3.7236 

3.7237 

3.7238 

3.7239 

37239 

3.7240 

3.7241  3.7242 

28  20 

3.7243 

3.7244 

3.7244 

3.7245 

3.7246 

3.7247 

3.7248 

3.7248 

3.7249!  3.7250 

28  30 

3.7251 

3.7252 

3.7253 

3.7253 

3.7254 

3.7255 

3.7256 

3.7257 

3.7257  3.7258 

28  40 

3.7259 

3.7260 

3.7261 

3.7262 

3.7262 

3.7263 

3.7264 

3.7265 

3.7266  3.7266 

28  50 

3.7267 

3.7268 

3.7269 

3.7270 

3.7271 

3.7271 

3.7272 

37273 

3.7274  3.7275 

1  29   0 

3.7275 

3.7276 

3.7277 

3.7278 

3.7279 

3.7279 

3.7280 

3.7281 

3.7282  3.7283 

29  10 

3.7284 

3.7284 

3.7285 

3.7286 

3.7287 

3.7288 

3.7288 

3.7289 

3.7290 

3.7291 

29  20 

3.7292 

3.7292 

3.7293 

3.7294 

3.7295 

3.7296 

3.7297 

3.7297 

3.7298 

3.7299 

29  30 

3.7300 

3.7301 

3.7301 

3.7302 

3.7303 

3.7304 

3.7305 

3.7305 

3.7306  3.7307 

29  40 

3.7308 

3.7309 

3.7309 

3.7310 

3.7311 

3.7312 

3.7313 

3.7313 

3.7314  3.7315 

29  50 

3.7316 

3.7317 

3.7317 

3.7318 

3.7319 

3.7320 

3.7321 

3.7322 

3.7322  !  3.7323 

TABLE    X. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

,-   Arc. 

d 

it 

2 

3 

</ 

4 

5 

6 

• 

7 

8 

9 

"o   /   « 
lh-30m-  0*- 

3.7324 

3.7325 

3.7326 

3.7326 

3.7327 

3.7328 

3.7329 

3.7330 

3.7330 

3.7331  ! 

30  10 

3.7332 

3.7333 

3.7334 

3.7334 

3.7335 

3.7336 

3.7337 

3.7338 

3.7338 

3.7339  1 

30  20 

3.7340 

3.7341 

3.7342 

3.7342 

3.7343 

3.7344 

3.7345 

3.7346 

3.7346 

3.7347 

30  30 

3.7348 

3.7349 

3.7350 

3.7350 

3.7351 

3.7352 

3.7353 

3.7354 

3.7354 

3.7355 

30  40 

3.7356 

3.7357 

3.7358 

3.7358 

3.7359 

3.7360 

3.7361 

3.7362 

3.7362 

3.7363 

30  50 

3.7364 

3.7365 

3.7366 

3.7366 

3.7367 

3.7368 

3.7369 

3.7370 

3.7370 

3.7371 

1  31   0 

3.7372 

3.7373 

3.7374 

3.7374 

3.7375 

3.7376 

3.7377 

3.7377 

3.7378 

3.7379 

31  10 

3.7380 

3.7381 

3.7381 

3.7382 

3.7383 

3.7384 

3.7385 

3.7385 

3.7386 

3.7387 

31  20 

3.7388 

3.7389 

3.7389 

3.7390 

3.7391 

3.7392 

3.7393 

3.7393 

3.7394  j  3.7395 

31  30 

3.7396 

3.7397 

3.7397 

3.7398 

3.7399 

3.7400 

3.7400 

3.7401 

3.7402 

3.7403  f 

31  40 

3.7404 

3.7404 

3.7405 

3.7406 

3.7407 

3.7408 

3.7408 

3.7409 

3.7410 

3.7411 

31  50 

3.7412 

3.7412 

3.7413 

3.7414 

3.7415 

3.7415 

3.7416 

3.7417 

3.7418 

3.7419 

1  32   0 

3.7419 

3.7420 

3.7421 

3.7422 

3.7423 

3.7423 

3.7424 

3.7425 

3.7426 

3.7426 

32  10 

3.7427 

3.7428 

3.7429 

3.7430 

3.7430 

3.7431 

3.7432 

3.7433 

3.7434 

3.7434 

32  20 

3.7435 

3.7436 

3.7437 

3.7437 

3.7438 

3.7439 

3.7440 

3.7441 

3.7441 

3.7442  ! 

32  30 

3.7443 

3.7444 

3.7444 

3.7445 

3.7446 

3.7447 

3.7448 

3.7448 

3.7449 

3.7450  : 

32  40 

3.7451 

3.7452 

3.7452 

3.7453 

3.7454 

3.7455 

3.7455 

3.7456 

3.7457  j  3.7458  " 

32  50 

3.7459 

3.7459 

3.7460 

3.7461 

3.7462 

3.7462 

3.7463 

3.7464 

3.7465 

3.7466 

1  33   0 

3.7466 

3.7467 

37468 

3.7469 

3.7469 

3.7470 

3.7471 

3.7472 

3.7473 

3.7473  5 

33  10 

3.7474 

3.7475 

3.7476 

3.7476 

3.7477 

3.7478 

3.7479 

3.7480 

3.7480 

3.7481 

33  20 

3.7482 

3.7483 

3.7483 

3.7484 

3.7485 

3.7486 

3.7487 

3.7487 

3.7488 

3.7489 

33  30 

3.7490 

3.7490 

3.7491 

3.7492 

3.7493 

3.7493 

3.7494 

3.7495 

3.7496 

3.7497 

33  40 

3.7497 

3.7498 

3.7499 

3.7500 

3.7500 

3.7501 

3.7502 

3.7503 

3.7504 

3.7504  | 

33  50 

3.7505 

3.7506 

3.7507 

3.7507 

3.7508 

3.7509 

3.7510 

3.7510 

3.7511 

3.7512 

1  34   0 

3.7513 

3.7514 

3.7514 

3.7515 

3.7516 

3.7517 

3.7517 

3.7518 

3.7519 

3.7520 

34  10 

3.7520 

3.7521 

3.7522 

3.7523 

3.7524 

3.7524 

3.7525 

3.7526 

3.75271  3.7527  j 

34  20 

3.7528 

3.7529 

3.7530 

3.7530 

3.7531 

3.7532 

3.7533 

3.7534 

3.7534  3.7535 

34  30 

3.7536 

3.7537 

3.7537 

3.7538 

3.7539 

3.7540 

3.7540 

3.7541 

3.7542  3.7543 

34  40 

3.7543 

3.7544 

3.7545 

3.7546 

3.7547 

3.7547 

3.7548 

3.7549 

3.7550  3.7550 

34  50 

3.7551 

3.7552 

3.7553 

3.7553 

3.7554 

3.7555 

3.7556 

3.7556 

3.7557,  3.7558 

1  35   0 

3.7559 

3.7560 

3.7560 

3.7561 

3.7562 

3.7563 

3.7563 

3.7564 

3.7565  1  3.7566 

35  10 

3.7566 

3.7567 

3.7568 

3.7569 

3.7569 

3.7570 

3.7571 

3.7572  3.7572  3.7573 

35  20 

3.7574 

3.7575 

3.7575 

3.7576 

3.7577 

3.7578 

3.7579 

3.7579  3.7580 

3.7581 

35  30 

3.7582 

3.7582 

3.7583 

3.7584 

3.7585 

3.7585 

3.7586 

3.7587 

3.7588 

3.7588 

35  40 

3,7589 

3.7590 

3.7591 

3.7591 

3.7592 

3.7593 

3.7594 

3.7594 

3.7595 

3.7596 

35  50 

3.7597 

3.7597 

3.7598 

3.7599 

3.7600 

3.7600 

3.7601 

3.7602 

3.7603 

3.7603 

1  36   0 

3.7604 

3.7605 

3.7606 

3.7606 

3.7607 

3.7608 

3.7609 

3.760$ 

3.7610 

3.7611 

36  10 

3.7612 

3.7613 

3.7613 

3.7614 

3.7615 

3.7616 

3.7616 

3.7617 

3.7618 

3.7619 

36  20 

3.7619 

3.7620 

3.7621 

3.7622 

3.7622 

3.7623 

3.7624 

3.7625 

3.7625 

3.7626 

36  30 

3.7627 

3.7628 

3.7628 

3.7629 

3.7630 

3.7631 

3.7631 

3.7632 

3.7633 

3.7634 

36  40 

3.7634 

3.7635 

3.7636 

3.7637 

3.7637 

3.7638 

3.7639 

3.7640 

3.7640 

37641  | 

36  50 

3.7642 

3.7643 

3.7643 

3.7644 

3.7645 

3.7645 

3.7646 

3.7647 

3.7648 

3.7648 

1  37   o 

3.7649 

3.7650 

3.7651 

3.7651 

3.7652 

3.7653 

3.7654 

3.7654 

3.7655 

3.7656 

37  10 

3.7657 

3.7657 

3.7658 

3.7659 

3.7660 

3.7660 

3.7661 

3.7662 

3.7663 

3.7663 

37  2o 

3.7664 

3.7665 

3.7666 

3.7666 

3.7667 

3.7668 

3.7669 

3.7669 

3.7670 

3.7671 

37  30 

3.7672 

3.7672 

3.7673 

3.7674 

3.7675 

3.7675 

3.7676 

3.7677 

3.7677 

3.7678 

37  40 

3.7679 

3.7680 

3.7681 

3.7681 

3.7682 

3.7683 

3.7683 

3.7684 

3.7685 

3.7686 

37  50 

3.7686 

3.7687 

3.7688 

3.7689 

3.7689 

3.7690 

3.7691 

3.7692 

3.7692 

3.7693 

1  38   0 

3.7694 

3.7695 

3.7695 

3.7696 

3.7697 

3.7697 

3.7698 

3.7699 

3.7700 

3.7700 

38  10 

3.7701 

3.7702 

3.7703 

3.7703 

3.7704 

3.7705 

3.7706 

3.7706 

3.7707 

3.7708 

38  20 

3.7709 

3.7709 

3.7710 

3.7711 

3.7711 

3.7712 

3.7713 

3.7714 

3.7714 

3.7715 

38  30 

3.7716 

3.7717 

3.7717 

3.7718 

3.7719 

3.7720 

3.7720 

3.7721 

3.7722 

3.7722  i 

38  40 

3.7723 

3.7724 

3.7725 

3.7725 

3.7726 

3.7727 

3.7728 

3.7728 

3.7729 

3.7730  I1 

38  50 

3.7731 

3.7731 

3.7732 

3.7733 

3.7733 

3.7734 

3.7735 

3.7736 

3.7736 

3.7737 

1  39   0 

3.7738 

3.7739 

3.7739 

3.7740 

3.7741 

3.7742 

3.7742 

3.7743 

3.7744 

3.7744  ' 

39  10 

3.7745 

3.7746 

3.7747 

3.7747 

3.7748 

3.7749 

3.7750 

3.7750 

3.7751 

3.7752 

39  20 

3.7752  3.7753 

3.7754 

3.7755 

3.7755 

3.7756 

3.7757 

3.7758 

3.7758 

3.7759 

39  30 

3.7760  3.7760 

3.7761 

3.7762 

3.7763 

3.7763 

3.7764 

3.7765 

3.7766 

3.7766 

39  40 

3.7767  3.7768 

3.7768 

3.7769 

3.7770 

3.7771 

3.7771 

3.7772  3.7773 

3.7774 

39  50 

3.7774  3.77751  3.7776 

3.7776 

3.7777 

3.7778J  3.7779 

3.7779,  3.7780 

3.7781 

49 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

0 

i 

2 

3 

i 

5 

6 

n 

7 

8 

9 

lh-40in-  0s 

3.7782 

3.7782 

3.7783 

3.7784 

3.7784 

3.7785 

3.7786 

3.7787 

3.7787 

3.7788 

40  10 

3.7789 

3.7789 

3.7790 

3.7791 

3.7792 

3.7792 

3.7793 

3.7794 

3.7795 

3.7795 

40  20 

3.7796 

3.7797 

3.7797 

3.7798 

3.7799 

3.7800 

3.7800 

3.7801 

3.7802 

3.7802 

40  30 

3.7803 

3.7804 

3.7805 

3.7805 

3.7806 

3.7807 

3.7807 

3.7808 

3.7809 

3.7810 

40  40 

3.7810 

3.7811 

3.7812 

3.7813 

3.7813 

3.7814 

3.7815 

3.7815 

3.7816 

3.7817 

40  50 

3.7818 

3.7818 

3.7819 

3.7820 

3.7820 

3.7821 

3.7822 

3.7823 

3.7823 

3.7824 

1  41   0 

3.7825 

3.7825 

3.7826 

3.7827 

3.7828 

3.7828 

3.7829 

3.7830 

3.7830 

3.7831 

41  10 

3.7832 

3.7833 

3.7833 

3.7834 

3.7835 

3.7835 

3.7836 

3.7837 

3.7838 

3.7838 

41  20 

3.7839 

3.7840 

3.7840 

3.7841 

3.7842 

3.7843 

3.7843 

3.7844 

3.7845 

3.7845 

41  30 

3.7846 

3.7847 

3.7848 

3.7848 

3.7849 

3.7850 

3.7850 

3.7851 

3.7852 

3.7853 

41  40 

3.7853 

3.7854 

3.7855 

3.7855 

3.7856 

3.7857 

3.7858 

3.7858 

3.7859 

3.7860 

41  50 

3.7860 

3.7861 

3.7862 

37863 

3.7863 

3.7864 

3.7865 

3.7865 

3.7866 

3.7867 

1  42   0 

3.7868 

3.7868 

3.7869 

3.7870 

3.7870 

3.7871 

3.7872 

3.7872 

3.7873 

3.7874 

42  10 

3.7875 

3.7875 

3.7876 

3.7877 

3.7877 

3.7878 

3.7879 

3.7880 

3.7880 

3.7881 

42  20 

3.7882 

3.7882 

3.7883 

3.7884 

3.7885 

3.7885 

3.7886 

3.7887 

3.7887 

3.7888 

42  30 

3.7889 

3.7889 

3.7890 

3.7891 

3.7892 

3.7892 

3.7893 

3.7894 

3.7894 

3.7895 

42  40 

3.7896 

3.7897 

3.7897 

3.7898 

3.7899 

3.7899 

3.7900 

3.7901 

3.7901 

3.7902 

42  50 

3.7903 

3.7904 

3.7904 

3.7905 

3.7906 

3.7906 

3.7907 

3.7908 

3.7908 

3.7909 

1  43   0 

3.7910 

3.7911 

3.7911 

3.7912 

3.7913 

3.7913 

3.7914 

3.7915 

3.7916 

3.7916 

43  10 

3.7917 

3.7918 

3.7918 

3.7919 

3.7920 

3.7920 

3.7921 

3.7922 

3.7923 

3.7923 

43  20 

3.7924 

3.7925 

3.7925 

3.7926 

3.7927 

3.7927 

3.7928 

3.7929 

3.7930 

3.7930 

43  30 

3.7931 

3.7932 

3.7932 

3.7933 

3.7934 

3.7934 

3.7935 

3.7936 

3.7937 

3.7937 

43  40 

3.7938 

3.7939 

3.7939 

3.7940 

3.7941 

3.7941 

3.7942 

3.7943 

3.7943 

3.7944 

43  50 

3.7945 

3.7946 

3.7946 

3.7947 

3.7948 

3.7948 

3.7949 

3.7950 

3.7950 

3.7951 

1  44   0 

3.7952 

3.7953 

3.7953 

3.7954 

3.7955 

3.7955 

3.7956 

3.7957 

3.7957 

3.7958 

44  10 

3.7959 

3.7959 

3.7960 

3.7961 

3.7962 

3.7962 

3.7963 

3.7964 

3.7964 

3.7965 

44  20 

3.7966 

3.7966 

3.7967 

3.7968 

3.7969 

3.7969 

3.7970 

3.7971 

3.7971 

3.7972 

44  30 

3.7973 

3.7973 

3.7974 

3.7975 

3.7975 

3.7976 

3.7977 

3.7978 

3.7978 

3.7979 

44  40 

3.7980 

3.7980 

3.7981 

3.7982 

3.7982 

3.7983 

3.7984 

3.7984 

3.7985 

3.7986 

44  50 

3.7987 

3.7987 

3.7988 

3.7989 

3.7989 

3.7990 

3.7991 

3.7991 

3.7992 

3.7993 

1  45   0 

3.7993 

3.7994 

3.7995 

3.7995 

3.7996 

3.7997 

3.7998 

3.7998 

3.7999 

3.8000 

45  10 

3.8000 

3.8001 

3.8002 

3.8002 

3.8003 

3.8004 

3.8004 

3.8005 

3.8006 

3.8006 

45  20 

3.8007 

3.8008 

3.8009 

3.8009 

3.8010 

3.8011 

3.8011 

3.8012 

3.8013 

3.8013 

45  30 

3.8014 

3.8015 

3.8015 

3.8016 

3.8017 

3.8017 

3.8018 

3.8019 

3.8020 

3.8020 

45  40 

3.8021 

3.8022 

3.8022 

3.8023 

3.8024 

3.8024 

3.8025 

3.8026 

3.8026 

3.8027 

45  50 

3.8028 

3.8028 

3.8029 

3.8030 

3.8030 

3.8031 

3.8032 

3.8033 

3.8033 

3.8034 

1  46   0 

3.8035 

3.8035 

3.8036 

3.8036 

3.8037 

3.8038 

3.8039 

3.8039 

3.8040 

3.8041 

46  10 

3.8041 

3.8042 

3.8043 

3.8043 

3.8044 

3.8045 

3.8045 

3.8046 

3.8047 

3.8048 

46  20 

3.8048 

3.8049 

3.8050 

3.8050 

3.8051 

3.8052 

3.8052 

3.8053 

3.8054 

3.8054 

46  30 

3.8055 

3.8056 

3.8056 

3.8057 

3.8058 

3.8058 

3.8059 

3.8060 

3.8060 

3.8061 

46  40 

3.8062 

3.8062 

3,8063 

3.8064 

3.8065 

3.8065 

3.8066 

3.8067 

3.8067 

3.8068 

46  50 

3.8069 

3.8069 

3,8070 

3.8071 

3.8071 

3.8072 

3.8073 

3.8073 

3.8074 

3.8075 

1  47   0 

3.8075 

3.8076 

3.8077 

3.8077 

3.8078 

3.8079 

3.8079 

3.8080 

3.8081 

3.8081 

47  10 

3.8082 

3.8083 

3.8083 

3.8084 

3.8085 

3.8085 

3.8086 

3.8087 

3.8088 

3.8088 

47  20 

3.8089 

3.8090 

3.8090 

.3.8091 

3.8092 

3.8092 

3,8093 

3.8094 

3.8094 

3.8095 

47  30 

3.8096 

3.8096 

3.8097 

3.8098 

3.8098 

3.8099 

3.8099 

3.8100 

3.8101 

3.8102 

47  40 

3.8102 

3.8103 

3.6104 

3.8104 

3.8105 

3.8106 

3.8106 

3.8107 

3.8108 

3.8108 

47  50 

3.8109 

3.8110 

3,8110 

3.8111 

3.8112 

3.8112 

3.8113 

3.8114 

3.8114 

3.8115 

1  48   0 

3.8116 

3.81  1€ 

3.8117 

3.8118 

3.8118 

3.8119 

3.8120 

3.8120 

3.8121 

3.8122 

48  10 

3.8122 

3.8123 

3.8124 

3.8124 

3.8125 

3.8126 

3.8126 

3.8127 

3.8128 

3.8128 

48  20 

3.8129 

3.8130 

3.8130 

3.8131 

3.8132 

3.8132 

3.8133 

3.8134 

3.8134 

3.8135 

48  30 

3.8136 

3.8136 

3.8137 

3.8138 

3.8138 

3.8139 

3.8140 

3.8140 

3.8141 

3.8142  i 

48  40 

3.8142 

3.8143 

3.8144 

3.8144 

3.8145 

3.8146 

3.8146 

3.8147 

3.8148 

3.8148 

48  50 

3.8149 

3.8150 

3.8150 

3.8151 

3.8152 

3.8152 

3.8153 

3.8154 

3.8154 

3.8155 

1  49   0 

3.8156 

3.8156 

3.8157 

3.8158 

3.8158 

3.8159 

3.8160 

3.8160 

3.8161 

3.8162 

49  10 

3.8162 

3.8163 

3.8164 

3.8164 

3.8165 

3.8166 

3.8166 

3.8167 

3.8168 

3.8168 

49  20 

3.8169 

3.8170 

3.8170 

3.8171 

3.8172 

3.8172 

3.8173 

3.8174 

3.8174 

3.8175 

49  30 

3.8176 

3.8176 

3.8177 

3.8178 

3.8178 

3.8179 

3.8180 

3.8180 

3.8181 

3.8182 

49  40 

3.8182 

3.8183 

3.8184 

3.8184 

3.8185 

3.8185 

3.8186 

3.8187 

3.8188 

3.8188 

49  50 

3.8189 

3.8190 

3.8190 

3.8191 

3.8191 

3.8192 

3.8193 

3.8193 

3.8194 

3.8195 

50 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

0 

ii 

2 

3 

4 

5 

6 

7 

8 

9 

0,    1    II 

lh-50m-  0s- 

3.8195 

3.8196 

3.8197 

3.8197 

3.8198 

3.8199 

3.8199 

3.8200 

3.8201 

3.8201 

50  10 

3.8202 

3.8203 

3.8203 

3.8204 

3.8205 

3.8205 

3.8206 

3.8207 

3.8207 

3.8206 

50  20 

3.8209 

3.8209 

3.8210 

3.8211 

38211 

38212 

3.8213 

3.8213 

3.8214 

3.8214 

50  30 

3.8215 

3.8216 

3.8216 

3.8217 

3.8218 

3.8218 

3.8219 

3.8220 

3.8220 

3.8221 

50  40 

3.8222 

3.8222 

3.8223 

3.8224 

3.8224 

3.8225 

3.8226 

3.8226 

3.8227 

3.8228 

50  50 

3.8228 

3.8229 

3.8230 

3.8230 

3.8231 

3.8231 

3.8232 

3.8233 

3.8233 

3.8234 

1  51   0 

3.8235 

3.8235 

3.8236 

3.8237 

3.8237 

3.8238 

3.8239 

3.8239 

3.8240 

3.8241 

51  10 

3.8241 

3.8242 

3.8243 

3.8243 

3.8244 

3.8245 

3.8245!  3.8246 

3.8246 

3.8247 

51  20 

3.8248 

3.8248 

3.8249 

3.8250 

3.8250 

3.8251 

3.8252  i  3.8252 

3.8253 

3.8254 

51  30 

3.8254 

3.8255 

3.8256 

3.8256 

3.8257 

3.8258 

3.8258 

3.8259 

3.8259 

3.8260 

51  40 

3.8261 

3.8261 

3.8262 

3.8263 

3.8263 

3.8264 

3.8265 

3.8265 

3.8266 

3.8267 

51  50 

3.8267 

3.8268 

3.8269 

3.8269 

3.8270 

3.8270 

3.8271 

3.8272 

3.8272 

3.8273 

1  52   0 

3.8274 

3.8274 

3.8275 

3.8276 

3.8276 

3.8277 

3.8278 

3.8278 

3.8279 

3.8280 

52  10 

3.8280 

3.8281 

3.8281 

3.8282 

3.8283 

3.8283 

3.8284 

3.8285 

3.8285 

3.8286 

52  20 

3.8287 

3.8287 

3.8288 

3.8289 

3.8289 

3.8290 

3.8290 

3.8291 

3.8292 

3.8292 

52  30 

3.8293 

3.8294 

3.8294 

3.8295 

3.8296 

3.8296 

3.8297 

3.8298 

3.8298 

3.8299 

52  40 

3.8299 

3.8300 

3.8301 

3.8301 

3.8302 

3.8303 

3.8303 

3.8304 

3.8305 

3.8305 

52  50 

3.8306 

3.8307 

3.8307 

3.8308 

3.8308 

3.8309 

3.8310 

3.8310 

3.8311 

3.8312 

1  53   0 

3.8312 

3.8313 

3.8314 

3.8314 

3.8315 

3.8315 

3.8316 

3.8317 

3.8317 

3.8318 

53  10 

3.8319 

3.8319 

3.8320 

3.8321 

3.8321 

3.8322 

3.8323 

3.8323 

3.8324 

3.8324 

53  20 

3.8325 

3.8326 

38326 

3.8327 

3.8328 

3.8328 

3.8329 

3.8330 

3.8330 

3.8331 

53  30 

3.8331 

3.8332 

3.8333 

3.8333 

3.8334 

3.8335 

3.8335 

3.8336 

3.8337 

3.8337 

53  40 

3.8338 

3.8338 

3.8339 

3.8340 

3.8340 

3.8341 

3.8342 

3.8342 

3.8343 

3.8344 

53  50 

3.8344 

3.8345 

3.8345 

3.8346 

3.8347 

3.8347 

3.8348 

3.8349 

3.8349 

3.8350 

1  54   0 

3.8351 

3.8351 

3.8352 

3.8352 

3.8353 

3.8354 

3.8354 

3.8355 

3.8356 

3.8356 

54  10 

3.8357 

3.8358 

3.8358 

3.8359 

3.8359 

3.8360 

3.8361 

3.8361 

3.8362 

3.8363 

54  20 

3.8363 

3.8364 

3.8365 

3.8365 

3.8366 

3.8366 

3.8367 

3.8368 

3.8368 

3.8369 

54  30 

3.8370 

3.8370 

3.8371 

3.8371 

3.8372 

3.8373 

3.8373 

3.8374 

3.8375 

3.8375 

54  40 

3.8376 

3.8377 

3.8377 

3.8378 

3.8378 

3.8379 

3.8380 

3.8380 

3.8381 

3.8382 

54  50 

3.8382 

3.8383 

3.8383 

3.8384 

3.8385 

3.8385 

3.8386 

3.8387 

3.8387 

3.8388 

1  55   0 

3.8388 

3.8389 

3.8390 

3.8390 

3.8391 

3.8392 

3.8392 

3.8393 

3.8394 

3.8394 

55  10 

3.8395 

3.8395 

3.8396 

3.8397 

3.8397 

3.8398 

3.8399 

3.8399 

3.8400 

3.8400 

55  20 

3.8401 

3.8402 

3.8402 

3.8403 

3.8404 

3.8404 

3.8405 

3.8405 

3.8406 

3.8407 

55  30 

3.8407 

3.8408 

3.8409 

3.8409 

3.8410 

3.8410 

3.8411 

3.8412 

3.8412 

3.8413 

55  40 

3.8414 

3.8414 

3.8415 

3.8415 

3.8416 

3.8417 

3.8417 

3.8418 

3.8419 

3.8419 

55  50 

3.8420 

3.8420 

3.8421 

3.8422 

3.8422 

3.8423 

3.8424 

3.8424 

3.8425 

3.8425 

1  56   0 

3.8426 

3.8427 

3.8427 

3.8428 

3.8429 

3.8429 

3.8430 

3.8430 

3.8431 

3.8432 

56  10 

3.8432 

3.8433 

3.8434 

3.8434 

3.8435 

3.8435 

3.8436 

3.8437 

3.8437 

3.8438 

56  20 

3.8439 

3.8439 

3.8440 

3.8440 

3.8441 

3.8442 

3.8442 

3.8443 

3.8444 

3.8444 

56  30 

3.8445 

3.8445 

3.8446 

3.8447 

3.8447 

3.8448 

3.8448 

3.8449 

3.8450 

3.8450 

56  40 

3.8451 

3.8452 

3.8452 

3.8453 

3.8453 

3.8454 

3.8455 

3.8455 

3.8456 

3.8457 

56  50 

3.8457 

3.8458 

3.8458 

3.8459 

3.8460 

3.8460 

3.8461 

3.8462 

3.8462 

3.8463 

1  57   0 

3.8463 

3.8464 

3.8465 

3.8465 

3.8466 

3.8466 

3.8467 

3.8468 

3.8468 

3.8469 

57  10 

3.8470 

3.8470 

3.8471 

3.8471 

3.8472 

3.8473 

3.8473 

3.8474 

3.8474 

3.8475 

57  20 

3.8476 

3.8476 

3.8477 

3.8478 

3.8478 

3.8479 

3.8479 

3.8480 

3.8481 

3.8481 

57  30 

3.8482 

3.8483 

3.8483 

3.8484 

3.8484 

3.8485 

3.8486 

3.8486 

3.8487 

3.8487 

57  40 

3.8488 

3.8489 

3.8489 

3.8490 

3.8491 

3.8491 

3.8492 

3.8492 

3.8493 

3.8494 

57  50 

3.8494 

38495 

3.8495 

3.8496 

3.8497 

3.8497 

3.8498 

3.8499 

3.8499 

3.8500 

1  58   0 

3.8500 

3.8501 

3.8502 

3.8502 

3.8503 

3.8503 

3.8504 

3.8505 

3.8505 

3.8506 

58  10 

3.8506 

3.8507 

3.8508 

3.8508 

3.8509 

3.8510 

3.8510 

3.8511 

3.8511 

3.8512 

58  20 

3.8513 

3.8513 

3.8514 

3.8514 

3.8515 

3.8516 

3.8516 

3.8517 

3.8517 

3.8518 

58  30 

3.8519 

3.8519 

3.8520 

3.8521 

3.8521 

3.8522 

3.8522 

3.8523 

3.8524 

3.8524 

58  40 

3.8525 

3.8525 

3.8526 

3.8527 

3.8527 

3.8528 

3.8528 

3.8529 

3.8530!  3.8530 

58  50 

3.8531 

3.8532 

3.8532 

3.8533 

3.8533 

3.8534 

3.8535 

3.8535 

3.8536  3.8536 

1  59   0 

3.8537 

3.8538 

3.8538 

3.8539 

3.8539 

3.8540 

3.8541 

3.8541 

3.8542;  3.8542 

59  10 

3.8543 

3.8544 

3.8544 

3.8545 

3.8545 

3.8546 

3.8547 

3.8547 

3.8548  3.8549 

59  20 

3.8549 

3.8550 

3.8550 

3.8551 

3.8552 

3.8552 

3.8553 

3.8553 

3.85541  3.8555 

59  30 

3.8555 

3.8556 

3.8556 

3.8557 

3.8558 

3.8558 

3.8559 

3.8559 

3.8560  3.8561 

59  40 

3.8561 

3.8562 

38562 

3.8563 

3.8564 

3.8564  3.8565 

3.8565 

3.8566!  3.8567 

59  50 

3.8567 

3.8568 

3.8568 

3.8569 

3.8570 

3.8570  3.8571 

3.8572 

3.8572  3.8573 

51 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

0 

1 

2 

3 

n 

4 

H 

5 

6 

7 

M 

8 

9 

0.         /          II 

2h.  Om.  Os. 

3.8573 

3.8574 

3.8575 

3.8575 

3.8576 

3.8576 

3.8577 

3.8578 

3.8578 

3.8579 

0    10 

3.8579 

3.8580 

3.8581 

3.8581 

3.8582 

3.8582 

3.8583 

3.8584 

3.8584 

3.8585 

0    20 

3.8585 

3.8586 

3.8587 

3.8587 

3.8588 

3.8588 

3.8589 

3.8590 

3.8590 

3.8591 

0    30 

3.8591 

3.8592 

3.8593 

3.8593 

3.8594 

3.8594 

3.8595 

3.8596 

3.8596 

3.8597 

0    40 

3.8597 

3.8598 

3.8599 

3.8599 

3.8600 

3.8600 

3.8601 

3.8602 

3.8602 

3.8603 

0    50 

3.8603 

3.8604 

3.8605 

3.8605 

3.8606 

3.8606 

3.8607 

3.8608 

3.8608 

3.8609 

2             0 

3.8609 

3.8610 

3.8611 

3.8611 

3.8612 

3.8612 

3.8613 

3.8614 

38614 

3.8615 

10 

3.8615 

3.8616 

3.8617 

3.8617 

3.8618 

3.8618 

3.8619 

3.8620 

3.8620 

3.8621 

20 

3.8621 

3.8622 

3.8623 

3.8623 

3.8624 

3.8624 

3.8625 

3.8625 

3.8626 

3.8627 

30 

3.8627 

3.8628 

3.8628 

3.8629 

3.8630 

3.8630 

3.8631 

3.8631 

3.8632 

3.8633 

40 

3.8633 

3.8634 

3.8634 

3.8635 

3.8636 

3.8636 

3.8637 

3.8637 

3.8638 

3.8639 

50 

3.8639 

3.8640 

3.8640 

3.8641 

3.8642 

3.8642 

3.8643 

3.8643 

3.8644 

3.8645 

220 

3.8645 

3.8646 

3.8646 

3.8647 

3.8647 

3.8648 

3.8649 

3.8649 

3.8650 

3.8650 

2    10 

3.8651 

3.8652 

3.8652 

3.8653 

3.8653 

3.8654 

3.8655 

3.8655 

3.8656 

3.8656 

2    20 

3.8657 

3.8658 

3.8658 

3.8659 

3.8659 

3.8660 

3.8661 

3.8661 

3.8662 

3.8662 

2    30 

3.8663 

3.8663 

3.8664 

3.8665 

3.8665 

3.8666 

3.8666 

3.8667 

3.8668 

3.8668 

2    40 

3.8669 

3.8669 

3.8670 

3.8671 

3.8671 

3.8672 

3.8672 

3.8673 

3.8673 

3.8674 

2    50 

3.8675 

3.8675 

3.8676 

3.8676 

3.8677 

3.8678 

3.8678 

3.8679 

3.8679 

3.8680 

230 

3.8681 

3.8681 

3.8682 

3.8682 

3.8683 

3.8684 

3.8684 

3.8685 

3.8685 

3.8686 

3    10 

3.8686 

3.8687 

3.8688 

3.8688 

3.8689 

3.8689 

3.8690 

3.8691 

3.8691 

3.8692 

3    20 

3.8692 

3.8693 

3.8693 

3.8694 

3.8695 

3.8695 

3.8696 

3.8696 

3.8697 

3.8698 

3    30 

3.8698 

3.8699 

3.8699 

3.8700 

3.8701 

3.8701 

3.8702 

3.8702 

3.8703 

3.8703    i 

3    40 

3.8704 

3.8705 

3.8705 

3.8706 

3.8706 

3.8707 

3.8708 

3.8708 

3.8700 

3.8709    ; 

3    50 

3.8710 

38710 

3.8711 

3.8712 

3.8712 

3.8713 

3.8713 

3.8714 

3.8715 

3.8715    ! 

240 

3.8716 

3.8716 

3.8717 

3.8717 

3.8718 

3.8719 

3.8719 

3.8720 

3.8720 

3.8721 

4    10 

3.8722 

3.8722 

3.8723 

3.8723 

3.8724 

3.8724 

3.8725 

3.8726 

3.8726 

3.8727 

4    20 

3.8727 

3.8728 

3.8729 

3.8729 

3.8730 

3.8730 

3.8731 

3.8731 

3.8732 

3.8733 

4    30 

3.8733 

3.8734 

3.8734 

3.8735 

3.8736 

3.8736 

3.8737 

3.8737 

3.8738 

3.8738 

4    40 

3.8739 

3.8740 

3.8740 

3.8741 

3.8741 

3.8742 

3.8742 

3.8743 

3.8744 

3.8744 

4    50 

3.8745 

3.8745 

3.8746 

3.8747 

3.8747 

3.8748 

3.8748 

3.8749 

3.8749 

3.8750 

250 

3.8751  j   3.8751 

3.8752 

3.8752 

3.8753 

3.8754 

3.8754 

3.8755 

3.8755 

3.8756 

5    10 

3.8756 

3.8757 

3.8758 

3.8758 

3.8759 

3.8759 

3.8760 

3.8760 

3.8761 

3.8762 

5    20 

3.8762 

3.8763 

3.8763 

3.8764 

3.8764 

3.8765 

3.8766 

3.8766 

3.8767 

3.8767 

,          5    30 

3.8768 

3.8769 

3.8769 

3.8770 

3.8770 

3.8771 

3.8771 

3.8772 

3.8773 

3.8773 

5    40 

3.8774 

3.8774 

3.8775 

3.8775 

3.8776 

3.8777 

3.8777 

3.8778 

38778 

3.8779 

5    50 

3.8779 

3.8780 

3.8781 

3.8781 

3.8782 

3.8782 

3.8783 

3.8783 

3.8784 

3.8785  1 

260 

3.8785 

3.8786 

3.8786 

3.8787 

3.8788 

3.8788 

3.8789 

3.8789 

3.8790 

3.8790 

6    10 

3.8791 

3.8792 

3.8792 

3.8793 

3.8793 

3.8794 

3.8794 

3.8795 

3.8796 

3.8796 

6    20 

3.8797 

3.8797 

3.8798 

3.8798 

3.8799 

3.8800 

3.8800 

3.8801 

3.8801 

3.8802 

6    30 

3.8802 

3.8803 

3.8804 

3.8804 

3.8805 

3.8805 

3.8806 

3.8806 

3.8807 

3.8808 

6    40 

3.8808 

3.8809 

3.8809 

3.8810 

3.8810 

3.8811 

3.8812 

3.8812 

3.8813 

3.8813 

6    50 

3.8814 

3.8814 

3.8815 

3.8816 

3.8816 

3.8817 

3.8817 

3.8818 

3.8818 

3.8819 

270 

3.8820 

3.8820 

3.8821 

3.8821 

3.8822 

3.8822 

3.8823 

3.8824 

3.8824 

3.8825 

7    10 

3.8825 

3.8826 

3.8826 

3.8827 

3.8828 

3.8828 

3.8829 

3.8829 

3.8830 

3.8830 

7    20 

3.8831 

3.8832 

3.8832 

3.8833 

3.8833 

3.8834 

3.8834 

3.8835 

3.8835 

3.8836 

7    30 

3.8837 

3.8837 

3.8838 

3.8838 

3.8839 

3.8839 

3.8840 

3.8841 

3.8841 

3.8842 

7    40 

3.8842 

3.8843 

3.8843 

3.8844 

3.8845 

3.8845 

3.8846 

3.8846 

3.8847 

3.8847 

7    50 

3.8848 

3,8849 

3.8849 

3.8850 

3.8850 

3.8851 

3.8851 

3.8852 

3.8852 

3.8853 

280 

3.8854 

3.8854 

3.8855 

3.8855 

3.8856 

3.8856 

3.8857 

3.8858 

3.8858 

3.8859 

8    10 

3.8859 

3.8860 

3.8860 

3.8861 

3.8862 

3.8862 

3.8863 

3.8863 

3.8864 

3.8864 

8    20 

3.8865 

3.8865 

3.8866 

3.8867 

a  8867 

3.8868 

3.8868 

3.8869 

3.8869 

3.8870 

8    30 

3.8871 

3.8871 

3.8872 

3.8872 

3  8873 

3.8873 

3.8874 

3.8874 

3.8875 

3.8876 

8    40 

3.8876 

3.8877 

3.8877 

3.8878 

3.8878 

3.8879 

3.8880 

3.8880 

3.8881 

3.8881 

8    50 

3.8882 

3.8882 

3.8883 

3.8883 

3.8884 

3.8885 

3.8885 

3.8886 

3.8886 

3.8887 

290 

38887 

3.8888 

3.8889 

3.8889 

3.8890 

3.8890 

3.8891 

3.8891 

3.8892 

3.8892 

9    10 

3.8893 

3.8894 

3.8894 

3.8895 

3.8895 

3.8896 

3.8896 

3.8897 

3.8897 

3.8898 

9    20 

3.8899 

3.8899 

3.8900 

3.8900 

3.8901 

3.8901 

3.8902 

3.8903 

3.8903 

3.8904 

9    30 

3.8904 

3.8905 

3.8905 

3.8906 

3.8906 

3.8907 

3.8908 

3.8908 

3.8909 

3.8909 

9    40 

3.8910 

3.8910!   38911 

3.8911 

3.8912 

3.8912 

3.8913 

3.8914 

3.8914 

3.8915 

9   50 

3.8915 

3.8916]   3.8916 

3.8917 

3.8918 

3.8918     3.8919 

3.8919 

3.8920 

3.8920 

52 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arp. 

0 

i 

2 

3 

it 

4 

5 

6 

7 

8 

H 

9 

2h-iom-  o3- 

3.8921 

3.8922 

3.8922 

3.8923 

3.8923 

3.8924 

3.8924 

3.8925 

3.8925 

3.8926 

10  10 

3.8927  j  3.8927 

3.8928 

3.8928 

3.8929 

3.8929 

3.8930 

3.8930 

3.8931 

3.8932 

10  20 

3.8932  3.8933 

3.8933 

3.8934 

3.8934 

3.8935 

3.8935 

3.8936 

3.8937 

3.8937 

10  30 

3.8938  3.8938 

3.8939 

3.8939 

3.8940 

3.8940 

3.8941 

3.8941 

3.8942 

3.8943 

10  40 

3.8943  3.8944 

3.8944 

3.8945 

3.8945 

3.8946 

3.8946 

3.8947 

3.8948 

3.8948 

10  50 

3.8949 

3.8949 

3.8950 

3.8950 

3.8951 

3.8951 

3.8952 

3.8953 

3.8953 

3.8954 

2  11   0 

3.8954 

3.8955 

3.8955 

3.8956 

3.8956 

3.8957 

3.8958 

3.8958 

3.8959 

3.8959 

11  10 

3.8960  1  3.8960 

3.8961 

3.8961 

3.8962 

3.8963 

3.8963 

3.8964 

3.8964 

3.8965 

11  20 

3.8965  3.8966 

3.8966 

3.8967 

3.8967 

3.8968 

3.8969 

3.8969 

3.8970 

3.8970 

11  30 

3.8971  3.8971 

3.8972 

3.8972 

3.8973 

3.8974 

3.8974 

3.8975 

3.8975 

3.8976 

11  40 

3.8976 

3.8977 

3.8977 

3.8978 

3.8978 

3.8979 

3.8980 

3.8980 

3.8981 

3.8981 

11  50 

3.8982 

3.8982 

3.8983 

3.8983 

3.8984 

3.8985 

3.8985 

3.8986 

3.8986 

3.8987 

2  12   0 

3.8987 

3.8988 

3.8988 

3.8989 

3.8989 

3.8990 

3.8991 

3.8991 

3.8992 

3.8992 

12  10 

3.8993 

3.8993 

3.8994 

3.8994 

3.8995 

3.8995 

3.8996 

3.8997 

3.8997 

3.8998 

12  20 

3.8998 

3.8999 

3.8999 

3.9000 

3.9000 

3.9001 

3.9001 

3.9002 

3.9003 

39003 

12  30 

3.9004 

3.9004 

3.9005 

3.9005 

3.9006 

3.9006 

3.9007 

3.9007 

3.9008 

3.9009 

12  40 

3.9009 

3.9010 

3.9010 

3.9011 

3.9011 

3.9012 

3.9012 

3.90131  3.9013 

3.9014 

12  50 

3.9015 

3.9015 

3.9016 

3.9016 

3.9017 

3.9017 

3.9018 

3.9018  3.9019 

3.9019 

2  13   0 

3.9020 

3.9021 

3.9021 

3.9022 

3.9022 

3.9023 

3.9023 

3.90241  3.9024 

3.9025 

13  10 

3.9025 

3.9026 

3.9027 

3.9027 

3.9028 

3.9028 

3.9029 

3.9029  3.9030 

3.9030 

13  20 

3.9031 

3.9031 

3.9032 

3.9033 

3.9033 

3.9034 

3.9034 

3.9035 

3.9035 

3.9036 

13  30 

3.9036 

3.9037 

3.9037 

3.9038 

3.9038 

3.9039 

3.9040 

39040 

3.9041 

3.9041 

13  40 

3.9042 

3.9042 

3.9043 

3.9043 

3.9044 

3.9044 

3.9045 

3.9046 

3.9046 

3.9047 

13  50 

3.9047 

3.9048 

3.9048 

3.9049 

3.9049 

3.9050 

3.9050 

3.9051 

3.9051 

3.9052 

2  14   0 

3.9053 

3.9053 

3.9054 

3.9054 

3.9055 

3.9055 

3.9056 

3.9056 

3.9057 

3.9057 

14  10 

3.9058 

3.9058 

3.9059 

3.9060 

3.9060 

3.9061 

3.9061 

3.9062 

3.9062 

3.9063 

14  20 

3.9063 

3.9064 

3.9064 

3.9065 

3.9066 

3.9066 

3.9067 

3.9067 

3.9068 

3.9068 

14  30 

3.9069 

3.9069 

3.9070 

3.9070 

3.9071 

3.9071 

3.9072 

3.9073 

3.9073 

3.9074 

14  40 

3.9074 

3.9075 

3.9075 

3.9076 

3.9076 

3.9077 

3.9077 

3.9078 

3.9078 

3.9079 

14  50 

3.9079 

3.9080 

3.9081 

3.9081 

3.9082 

39082 

3.9083 

3.9083 

3.9084 

3.9084 

2  15   0 

3.9085 

3.9085 

3.9086 

3.9086 

3.9087 

3.9088 

3.9088 

3.9089 

3.9089 

3.9090 

15  10 

3.9090 

3.9091 

3.9091 

3.9092 

3.9092 

3.9093 

3.9093 

3.9094 

3.9094 

3.9095 

15  20 

3.9096 

3.9096 

3.9097 

3.9097 

3.9098 

3.9098 

3.9099 

3.9099 

3.9100 

3.9100 

15  30 

3.9101 

3.9101 

3.9102 

3.9103 

3.9103 

3.9104 

3.9104 

3.9105 

3.9105 

3.9106 

15  40 

3.9106 

3.9107 

3.9107 

3.9108 

3.9108 

3.9109 

3.9109 

3.9110 

3.9111 

3.9111 

15  50 

3.9112 

3.9112 

3.9113 

3.9113 

3.9114 

3.9114 

3.9115 

3.9115 

3.9116 

3.9116 

1  2  16   0 

3.9117 

3.9117 

3.9118 

3.9118 

3.9119 

3.9120 

3.9120 

3.9121 

3.9121 

3.9122 

16  10 

3.9122 

3.9123 

3.9123 

3.9124 

3.9124 

3.9125 

3.9125 

3.9126 

3.9126 

3.9127 

16  20 

3.9128 

3.9128 

3.9129 

3.9129 

3.9130 

3.9130 

3.9131 

3.9131 

3.9132 

3.9132 

16  30 

3.9133 

3.9133 

3.9134 

3.9134 

3.9135 

3.9135 

3.9136 

3.9137 

3.9137 

3.9138 

16  40 

3.9138 

3.9139 

3.9139 

3.9140 

3.9140 

3.9141 

3.9141 

3.9142 

3.9142 

3.9143 

16  50 

3.9143 

3.9144 

3.9144 

3.9145 

3.9146 

3.9146 

3.9147 

3.9147 

3.9148 

3.9148 

2  17   0 

3.9149 

3.9149 

3.9150 

3.9150 

3.9151 

3.9151 

3.9152 

3.9152 

3.9153 

3.9153 

17  10 

3.9154 

3.9155 

3.9155 

3.9156 

3.9156 

3.9157 

3.9157 

3.9158 

3.9158 

3.9159  l! 

17  20 

3.9159 

3.9160 

3.9160 

3.9161 

3.9161 

3.9162 

3.9162 

3.9163 

3.9163 

3.9164  j 

17  30 

3.9165 

3.9165 

3.9166 

3.9166 

3.9167 

3.9167 

3.9168 

3.9168 

3.9169 

3.9169  { 

17  40 

3.9170 

3.9170 

3.9171 

3.9171 

3.9172 

3.9172 

3.9173 

3.9173 

3.9174 

3.9175  !• 

17  50 

3.9175 

3.9176 

3.9176 

3.9177 

3.9177 

3.9178 

3.9178 

3.9179 

3.9179 

3.9180 

2  18   0 

3.9180 

3.9181 

3.9181 

3.9182 

3.9182 

3.9183 

3.9183 

3.9184 

3.9184 

3.9185 

18  10 

3.9186 

3.9186 

3.9187 

3.9187 

3.9188 

3.9188 

3.9189 

3.9189 

3.9190 

3.9190 

18  20 

3.9191 

3.9191 

3.9192 

3.9192 

3.9193 

3.9193 

3.9194 

3.9194 

3.9195 

3.9195 

18  30 

3.9196 

3.9197 

3.9197 

3.9198 

3.9198 

3.9199 

3.9199 

3.9200 

3.9200 

3.9201 

18  40 

3.9201 

3.9202 

3.9202 

3.9203 

3.9203 

3.9204 

3.9204 

3.9205 

3.9205 

3.9206 

18  50, 

3.9206 

3.9207 

3.9207  3.9208 

3.9209 

3.9209 

3.9210 

3.9210 

3.9211 

3.9211 

2  19   0 

3.9212 

3.9212 

3.9213  3.9213 

3.9214 

3.9214 

3.9215 

3.9215 

3.9216 

3.9216 

19  10 

3.9217  3.9217 

3.9218 

3.9218 

3.9219 

3.9219 

3.9220 

3.9221 

3.9221 

3.9222 

19  20 

3.9222  3.9223 

3.9223 

3.9224 

3.9224 

3.9225 

3.9225 

3.9226 

3.9226 

3.9227 

19  30 

3.9227  3.9228 

3.9228 

3.9229 

3.9229 

3.9230  3.9230 

3.9231 

3.9231 

3.9232 

19  40 

3.92321  3.9233 

3.9233  3.9234 

3.9235 

3.9235  3.9236 

3.9236 

3.9237 

3.9237 

19  50 

3.9238  ''  3.9238  !  3.9239  3.9239  3.9240 

3.9240  3.9241 

3.9241 

3.9242  j  3.9242 

53 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

d 

i 

H 

2 

3 

i 

5 

6 

7 

8" 

9 

0.    I 

2h-20m-  0s- 

3.9243 

3.9243 

3.9244 

3.9244 

3.9245 

3.9245 

3.9246 

3.9246 

3.9247 

39247 

20  10 

3.9248 

3.9248 

3.9249 

3.9250 

3.9250 

3.9251 

3.9251 

3.9252 

3.9252i 

3.9253 

20  20 

3.9253 

3.9254 

3.9254 

3.9255 

3.9255 

3.9256 

3.9256 

3.9257 

3.9257 

3.9258 

20  30 

3.9258 

3.92§9 

3.9259 

3.9260 

3.9260 

3.9261 

3.9261 

3.9262 

3.9262 

3.9263 

20  40 

3.9263 

3.9264 

3.9264 

3.9265 

3.9265 

3.9266 

3.9267 

3.9267 

3.9268 

3.9268 

20  50 

3.9269 

3.9269 

3.9270 

3.9270 

3.9271 

3.9271 

3.9272 

3.9272 

3.9273 

3.9273 

2  21   0 

3.9274 

3.9274 

3.9275 

3.9275 

3.9276 

3.9276 

3.9277 

3.9277 

3.9278 

3.9278 

21  10 

3.9279 

3.9279  3.9280 

3.9280 

3.9281 

3.9281 

3.9282 

3.9282 

3.9283 

3.9283 

21  20 

3.9284 

3.9284  3.9285 

3.9285 

3.9286 

3.9287 

3.9287 

3.9288 

3.9288 

3.9289 

21  30 

3.9289 

3.9290  3.9290 

3.9291 

3.9291 

3.9292 

3.9292 

3.9293 

3.9293 

3.9294 

21  40 

3.9294 

3.9295 

3.9295 

3.9296 

3.9296 

3.9297 

3.9297 

3.9298 

3.9298 

3.9299 

21  50 

3.9299 

3.9300 

3.9300 

3.9301 

3.9301 

3.9302 

3.9302 

3.9303 

3.9303 

3.9304 

2  22   0 

3.9304 

3.9305 

3.9305 

3.9306 

3.9306 

3.9307 

3.9307 

3.9308 

3.9308 

3.9309 

22  10 

3.9309 

3.9310 

3.9311 

3.9311 

3.9312 

3.9312 

3.9313 

3.9313 

3.9314 

3.9314 

22  20 

3.9315 

3.9315 

3.9316 

3.9316 

3.9317 

3.9317 

3.9318 

3.9318 

3.9319 

3.9319 

22  30 

3.9320 

3.9320 

3.9321 

3.9321 

3.9322 

3.9322 

3.9323 

3.9323 

3.9324 

3.9324 

22  40 

3.9325 

3.9325 

3.9326 

3.9326 

3.9327 

3.9327 

3.9328 

3.9328 

3.9329 

3.9329 

22  50 

3.9330 

3.9330 

3.9331 

3.9331 

3.9332 

3.9332 

3.9333 

3.9333 

3.9334 

3.9334 

2  23   0 

3.9335 

"  3.9335 

3.9336 

3.9336 

3.9337 

3.9337 

3.9338 

3.9338 

3.9339 

3.9339 

23  10 

3.9340 

3.9340 

3.9341 

3.9341 

3.9342 

3.9342 

3.9343 

3.9343 

3.9344 

3.9344 

23  20 

3.9345 

3.9345 

3.9346 

3.9346 

3.9347 

3.9348, 

3.9348 

3.9349 

3.9349 

3.9350 

23  30 

3.9350 

3.9351 

3.9351 

3.9352 

3.9352 

3.9353 

3.9353 

39354 

3.9354 

3.9355 

23  40 

3.9355 

3.9356 

3.9356 

3.9357 

3.9357 

3.9358 

3.9358 

3.9359 

3.9359 

3.9360 

23  50 

3.9360 

3.9361 

3.9361 

3.9362 

3.9362 

3.9363 

3.9363 

3.9364 

3.9364 

3.9365 

2  24   0 

3.9365 

3.9366 

3.9366 

3.9367 

3.9367 

3.9368 

3.9368 

3.9369 

3.9369 

3.9370 

24  10 

3.9370 

3.9371 

3.9371 

3.9372 

3.9372 

3.9373 

3.9373 

3.9374 

3.9374 

3.9375 

24  20 

3.9375 

3.9376 

3.9376 

3.9377 

3.9377 

3.9378 

3.9378 

3.9379 

3.9379 

3.9380 

24  30 

3.9380 

3.9381 

3.9381 

3.9382 

3.9382 

3.9383 

3.9383 

3.9384 

3.9384 

3.9385 

24  40 

3.9385 

3.9386 

3.9386 

3.9387 

3.9387 

3.9388 

3.9388 

3.9389 

3.9389 

3.9390 

24  50 

3.9390 

3.9391 

3.9391 

3.9392 

3.9392 

39393 

3.9393 

3.9394 

3.9394 

3.9395 

2  25   0 

3.9395 

3.9396 

3.9396 

3.9397 

3.9397 

3.9398 

3.9398 

3.9399 

3.9399 

3.9400 

25  10 

3.9400 

3.9401 

3.9401 

3.9402 

3.9402 

3.9403 

3.9403 

3.9404 

3.9404 

3.9405 

25  20 

3.9405 

3.9406 

3.9406 

3.9407 

3.9407 

3.9408 

3.9408 

3.9409 

3.9409 

3.9410 

25  30 

3.9410 

3.9411 

3.9411 

3.9412 

3.9412 

3.9413 

3.9413 

3.9414 

3.9414 

3.9415 

25  40 

3.9415 

3.9416 

3.9416 

3.9417 

3.9417 

3.9418 

3.9418 

3.9419 

3.9419 

3.9420 

25  50 

3.9420 

3.9421 

3.9421 

3.9422 

3.9422 

3.9423 

3.9423 

3.9424 

3.9424 

3.9425 

2  26   0 

3.9425 

3.9426 

3.9426 

3.9427 

3.9427 

3.9428 

3.9428 

3.9429 

3.9429 

3.9430 

26  10 

3.9430 

3.9430 

3.9431 

3.9431 

3.9432 

3.9432 

3.9433 

3.9433 

3.9434 

3.9434 

26  20 

3.9435 

3.9435 

3.9436 

3.9436 

3.9437 

3.9437 

3.9438 

3.9438 

3.9439 

3.9439 

26  30 

3.9440 

3.9440 

3.9441 

3.9441 

3.9442 

3.9442 

3.9443 

3.9443 

3.9444 

3.9444 

26  40 

3.9445 

3.9445 

3.9446 

3.9446 

3.9447 

3.9447 

3.9448 

3.9448 

3.9449 

3.9449 

26  50 

3.9450 

3.9450 

3.9451 

3.9451 

3.9452 

3.9452 

3.9453 

3.9453 

3.9454 

3.9454 

2  27   0 

3.9455 

3.9455 

3.9456 

3.9456 

3.9457 

3.9457 

3.9458 

3.9458 

3.9459 

3.9459 

27  10 

3.9460 

3.9460 

3.9461 

3.9461 

3.9462 

3.9462 

3.9463 

3.9463 

3.9464 

3.9464 

27  20 

3.9465 

3.9465 

3.9466 

3.9466 

3.9466 

3.9467 

3.9467 

3.9468 

3.9468 

3.9469 

27  30 

3.9469 

3.9470 

3.9470 

3.9471 

3.9471 

3.9472 

3.9472 

3.9473 

3.9473 

3.9474 

27  40 

3.9474 

3.9475 

3.9475 

3.9476 

3.9476 

3.9477 

3.9477 

3.9478 

3.9478 

3.9479 

27  50 

3.9479 

3.9480 

3.9480 

3.9481 

3.9481 

3.9482 

3.9482 

3.9483 

3.9483 

3.9484 

2  28   0 

3.9484 

3.9485 

3.9485 

3.9486 

3.9486 

3.9487 

3.9487 

3.9488 

3.9488 

3.9489 

28  10 

3.9489 

3.9490 

3.9490 

3.9490 

3.9491 

3.9491 

3.9492 

3.9492 

3.9493 

3.9493 

28  20 

3.9494 

3.9494 

3.9495 

3.9495 

3.9496 

3.9496 

3.9497 

3.9497 

3.9498 

3.9498 

28  30 

3.9499 

3.9499 

3.9500 

3.9500 

3.9501 

3.9501 

3.9502 

3.9502 

3.9503 

3.9503 

28  40 

3.9504 

3.9504 

3.9505 

3.9505 

3.9506 

3.9506 

3.9507 

3.9507 

3.9508 

3.9508 

28  50 

3.9509 

3.9509 

3.9509 

3.9510 

3.9510 

3.9511 

3.9511 

3.9512 

3.9512 

3.9513 

2  29   0 

3.9513 

3.9514 

3.9514  3.9515 

3.9515 

3.9516 

3.9516 

3.9517 

3.9517 

3.9518 

29  10 

3.9518 

3.9519 

3.951  9  j  3.9520 

3.9520 

3.9521 

3.9521 

3.9522 

3.9522 

3.9523 

29  20 

3.9523 

3.9524 

3.9524  3.9525 

3.9525 

3.9526 

3.9526 

3.9526 

3.9527 

3.9527 

29  30 

3.9528 

3.9528 

3.9529  3.9529 

3.9530 

3.S530 

3.9531 

3.9531 

3.9532 

3.9532 

29  40 

3.9533 

3.9533 

3.9534  3.9534 

3.9535 

3.9535 

3.9536 

3.9536 

3.9537 

3.9537 

29  50 

3.9538  3.9538 

3.9539  3.9539!  3.9540 

3.9540 

3.9540 

3.9541 

3.9541 

3.9542 

TABLE    15. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

d 

i 

2 

3 

• 

4 

& 

6 

7 

4 

§ 

0,    (    // 

2h.30m.  QS. 

3.9542 

3.9543 

3.9543 

3.9544 

3.9544 

3.9545 

3.9545 

3.9546 

3.9546 

3.9547 

30  10 

3.9547 

3.9548 

S.9548 

3.9549 

3.9549 

3.9550 

3.9550 

3.9551 

3.9551 

3.9552 

30  20 

3.9552 

3.9553 

3.9553 

3.9554 

3.9554 

3.9554 

3.9555 

3.9555 

3.9556 

3.9556 

30  30 

3.9557 

3.9557 

3.9558 

3.9558 

3.9559 

3.9559 

3.9560 

3.9560 

3.9561 

3.9561 

30  40 

3.9562 

3.9562 

3.9563 

3.9563 

3.9564 

3.9564 

3.9565 

3.9565 

3.9566 

3.9566 

30  50 

3.9566 

3.9567 

3.9567 

3.9568 

3.9568 

3.9569 

3.9569 

3.9570 

3.9570 

3.9571 

2  31   0 

3.9571 

3.9572 

3.9572 

3.9573 

3.9573 

3.9574 

3.9574 

3.9575 

3.9575 

3.9576 

31  10 

3.9576 

3.9577 

3.9577 

3.9578 

3.9578 

3.9578 

3.9579 

3.9579 

3.9580 

3.9580 

31  20 

3.9581 

3.9581 

3.9582 

3.9582 

3.9583 

3.9583 

3.9584 

3.9584 

39585 

3.9585 

31  30 

3.9586 

3.9586 

3.9587 

3.9587 

3.9588 

3.9588 

3.9589 

3.9589 

3.9589 

3.9590 

31  40 

3.9590 

3.9591 

3.9591 

3.9592 

3.9592 

3.9593 

3.9593 

3.9594 

3.9594 

3.9595 

31  50 

3.9595 

3.9596 

3.9596 

3.9697 

3.9597 

3.9598 

3.9598 

3.9599 

3.9599 

3.9599 

2  32   0 

3.9600 

3.9600 

3.9601 

3.9601 

3.9602 

3.9602 

3.9603 

3.9603 

3.9604 

3.9604 

32  10 

3.9605 

3.9605 

3.9606 

3.9606 

3.9607 

3.9607 

3.9608 

3.9608 

3.9609 

3.9609 

32  20 

3.9609 

3.9610 

3.9610 

3.9611 

3.9611 

3.9612 

3.9612 

3.9613 

3.9613 

3.9614 

32  30 

3.9614 

3.9615 

3.9615 

3.9616 

3.9616 

3.9617 

3.9617 

3.9618 

3.9618 

3.9618 

32  40 

3.9619 

3.9619 

3.9620 

3.9620 

3.9621 

3.9621 

3.9622 

3.9622 

3.9623 

39623 

32  50 

3.9624 

3.9624 

3.9625 

3.9625 

3.9626 

3.9626 

3.9627 

3.9627 

3.9627 

3.9628 

2  33   0 

3.9628 

3.9629 

3.9629 

3.9630 

3.9630 

3.9631 

3.9631 

3.9632 

3.9632 

3.9633 

33  10 

3.9633 

3.9634 

3.9634 

3.9634 

3.9635 

3.9635 

3.9636 

3.9636 

3.9637 

3.9637 

33  20 

3.9638 

3.9638 

3.9639 

3.9639 

3.9640 

3.9640 

3.9641 

3.9641 

3.9642 

3.9642 

33  30 

3.9642 

3.9643 

3.9643 

3.9644 

3.9644 

3.9645 

3.9645 

3.9646 

3.9646 

3.9647 

33  40 

3.9647 

3.9648 

3.9648 

3.9649 

3.9649 

3.9650 

3.9650 

3.9651 

3.9651 

3.9652 

33  50 

3.9652 

39653 

3.9653 

3.9653 

3.9654 

3.9654 

3.9655 

3.9655 

3.9656 

3.9656 

2  34   0 

3.9657 

3.9657 

3.9658 

3.9658 

3.9658 

3.9659 

3.9659 

3.9660 

3.9660 

3.9661 

34  10 

3.9661 

3.9662 

3.9662 

3.9663 

3.9663 

3.9664 

3.9664 

3.9665 

3.9665 

3.9665 

34  20 

3.9666 

3.9666 

3.9667 

3.9667 

3.9668 

3.9668 

3.9669 

3.9669 

3.9670 

3.9670 

34  30 

3.9671 

3.9671 

3.9672 

3.9672 

3.9672 

3.9673 

3.9673 

3.9674 

3.9674 

3.9675 

34  40 

3.9675 

3.9676 

3.9676 

3.9677 

3.9677 

3.9678 

3.9678 

3.9679 

3.9679 

3.9680 

34  50 

3.9680 

3.9681 

3.9681 

3.9682 

3.9682 

3.9682 

3.9683 

3.9683 

3.9684 

3.9684 

2  35   0 

3.9685 

3.9685 

3.9686 

3.9686 

3.9687 

3.9687 

3.9688 

3.9688 

3.9689 

3.9689 

35  10 

3.9689 

3.9690 

3.9690 

3.9691 

3.9691 

3.9692 

3.9692 

3.9693 

3.9693 

3.9694 

35  20 

3.9694 

3.9695 

3.9695 

3.9696 

3.9696 

3.9696 

3.9697 

3.9697 

3.9698 

3.9698 

35  30 

3.9699 

3.9699 

3.9700 

3.9700 

3.9701 

3.9701 

3.9702 

3.9702 

3.9703 

3.9703 

35  40 

3.9703 

3.9704 

3.9704 

3.9705 

3.9705 

3.9706 

3.9706 

3.9707 

3.9707 

3.9708 

35  50 

3.9708 

3.9709 

3.9709 

3.9710 

3.9710 

3.9710 

3.9711 

3.9711 

3.9712 

3.9712 

2  36   0 

3.9713 

3.9713 

3.9714 

3.9714 

3.9715 

3.9715 

3.974C 

3.9716 

3.9716 

3.9717 

36  10 

3.9717 

3.9718 

3.9718 

3.9719 

3.9719 

3.9720 

3.9720 

3.9721 

3.9721 

3.9722 

36  20 

3.9722 

3.9722 

3.9723 

3.9723 

3.9724 

3.9724 

3.9725 

3.9725 

3.9726 

3.9726 

36  30 

3.9727 

3.9727 

3.9728 

3.9728 

3.9729 

3.9729 

3.9729 

3.9730 

3.9730 

3.9731 

36  40 

3.9731 

3.9732 

3.9732 

3.9733 

3.9733 

3.9734 

3.9734 

3.9735 

3.9735 

3.9735 

36  50 

3.9736 

3.9736 

3.9737 

3.9737 

3.9738 

3.9738 

3.9739 

3.9739 

3.9740 

3.9740 

2  37   0 

3.9741 

3.9741 

3.9741 

3.9742 

3.9742 

3.9743 

3.9743 

3.9744 

3.9744 

3.9745 

37  10 

3.9745 

3.9746 

39746 

3.9746 

3.9747 

3.9747 

3.9748 

3.9748 

3.9749 

3.9749 

37  20 

3.9750 

3.9750 

3.9751 

3.9751 

3.9752 

3.9752 

3.9752 

3.9753 

3.9753 

3.9754 

37  30 

3.9754 

3.9755 

3.9755 

3.9756 

3.9756 

3.9757 

3.9757 

3.9758 

3.9758 

3.9758 

37  40 

3.9759 

3,9759 

3.9760 

3.9760 

3.9761 

3.9761 

3.9762 

3.9762 

3.9763 

3.9763 

37  50 

3.9763 

39764 

3.9764 

3.9765 

3.9765 

3.9766 

3.9766 

3.9767 

3.9767 

3.9768 

2  38   0 

3.9768 

3.9769 

3.9769 

3.9769 

3.9770 

3.9770 

3.9771 

3.9771 

3.9772 

3.9772 

38  10 

3.9773 

3.9773 

3.9774 

3.9774 

3.9774 

3.9775  3.9775 

3.9776 

3.9776 

3.9777 

38  20 

3.9777 

3.9778 

3.9778 

3.0779 

3.9779 

3.9779  3.9780 

3.9780 

3.9781 

3.9781 

38  30 

3.9782 

3.9782 

3.9783 

3.9783 

3.9784 

3.9784  1  3.9785 

3.9785 

3.9785 

3.9786 

38  40 

3.9786 

3.9787 

3.9787 

3.9788 

3.9788 

3.9789  3.9889 

3.9790 

3.9790 

3.9790 

38  50 

3.9791 

3.9791 

3.9792 

3.9792 

3.9793 

3.9793  3.9794 

3.9794 

3.9795 

3.9795 

2  39   0 

3.9795 

3.9796 

3.9796 

3.9797 

3.9797 

3.9798  3.9798 

3.9799 

3.9799 

3.9800 

39  10 

3.9800 

3.9800 

3.9801 

3.9801 

3.9802 

3.9802  i  3.9803 

3.9803 

3.9804 

3.9804 

39  20 

3.9805 

3.9805 

3.9805 

3.9806 

3.9806 

3.98071  3.9807 

3.9808 

3.9808 

3.9809 

39  30 

3.9809 

3.9810 

3.9810 

3.9810 

3.9811 

3.981  11  3.9812 

3.9812 

3.9813 

3.98ia 

39  40 

3.9814 

3.9814 

3.9815 

3.9815 

3.9815 

3.9816  3.9816 

3.9817 

3.9817 

3.9818 

39  50 

3.9818 

3.9819 

3.9S19 

3.9819 

3.9820 

3.9820  3.9821 

3.9821 

3.9822  3.9822 

55 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

0 

i 

ii 

2 

3 

4 

5 

n 

6 

• 

8" 

9 

O.    1    11 

2h.40m.  £. 

3.9823 

3.9823 

3.9824 

3.9824 

3.9825 

3.9825 

3.9825 

3.9826 

3.9826 

3.9827 

40  10 

3.9827 

3.9828 

3.9828 

3.9829 

3.9829 

3.9829 

3.9830 

3.9830 

3.9831 

3.9831 

40  20 

3.9832 

3.9832 

3.9833 

3.9833 

3.9834 

3.9834 

3.9834 

3.9835 

3.9835 

3.9836 

40  30 

3.9836 

3.9837 

3.9837 

3.9838 

3.9838 

3.9839 

3.9839 

3.9839 

3.9840 

3.9840 

40  40 

3.9841 

3.9841 

3.9842 

3.9842 

3.9843 

3.9843 

3.9843 

3.9844 

3.9844 

3.9845 

40  50 

3.9845 

3.9846 

3.9846 

3.9^47 

3.9847 

3.9848 

3.9848 

3.9848 

3.9849 

3.9849 

2  41   0 

3.9850 

3.9850 

3.9851 

3.9851 

3.9852 

3.9852 

3.9852 

3.9853 

3.9853 

3.9854 

41  10 

3.9854 

3.9855 

3.9855 

3.9856 

3.9856 

3.9857 

3.9857 

3.9857 

3.9858 

3.9858 

41  20 

3.9859 

3.9859 

3.9860 

3.9860 

3.9861 

3.9861 

3.9861 

3.9862 

3.9862 

3.9863 

41  30 

3.9863 

3.9864 

3.9864 

3.9865 

3.9865 

3.9865 

3.9866 

3.9866 

3.9867 

3.9867 

41  40 

3.9868 

3.9868 

3.9869 

3.9869 

3.9870 

3.9870 

3.9870 

3.9871 

3.9871 

3.9872 

41  50 

3.9872 

3.9873 

3.9873 

3.9874 

3.9874 

3.9874 

3.9875 

3.9875 

3.9876 

3.9876 

2  42   0 

3.9877 

3.9877 

3.9878 

3.9878 

3.9878 

3.9879 

3.9879 

3.9880 

3.9880 

3.9881 

42  10 

3.9881 

3.9882 

3.9882 

3.9882 

3.9883 

3.9883 

3.9884 

3.9884 

3.9885 

3.9885 

42  20 

3.9886 

3.9886 

3.9886 

3.9887 

3.9887 

3.9888 

3.9888 

3.9889 

3.9889 

3.9890 

42  30 

3.9890 

3.9890 

3.9891 

3.9891 

3.9892 

3.9892 

3.9893 

3.9893  3.9894 

3.9894 

42  40 

3.9894 

3.9895 

3.9895 

3.9896 

3.9896 

3.9897 

3.9897 

3.9898  3.9898 

3.9898 

42  50 

3.9899 

3.9899 

3.9900 

3.9900 

3.9901 

3.9901 

3.9902 

3.9902 

3.9903 

3.9903 

2  43   0 

3.9903 

3.9904 

3.9904 

3.9905 

3.9905 

3.9906 

3.9906 

3.9906 

3.9907 

3.9907 

43  10 

3.9908 

3.9908 

3.9909 

3.9909 

3.9910 

3.9910 

3.9910 

3.9911 

3.9911 

3.9912 

43  20 

3.9912 

3.9913 

3.9913 

3.9914 

3.9914 

3.9914 

3.9915 

3.9915 

3.9916 

3.9916 

43  30 

3.9917 

3.9917 

3.9918 

3.9918 

3.9918 

3.9919 

3.9919 

39920 

3.9920 

3.9921 

43  40 

3.9921 

3.9922 

3.9922 

3.9922 

3.9923 

3.9923 

3.9924 

3.9924  !  3.9925 

3.9925 

43  50 

3.9926 

3.9926 

3.9926 

3.9927 

3.9927 

3.9928 

3.9928 

3.9929 

3.9929 

3.9930 

2  44   0 

3.9930 

3.9930 

3.9931 

3.9931 

3.9932 

3.9932 

3.9933 

3.9933 

3.9933 

3.9934 

44  10 

3.9934 

3.9935 

3.9935 

3.9936 

3.9936 

3.9937 

3.9937 

3.9937 

3.9938 

3.9938 

44  20 

3.9939 

3.9939 

3.9940 

3.9940 

3.9941 

3.9941 

3.9941 

3.9942 

3.9942 

3.9943 

44  30 

3.9943 

3.9944 

3.9944 

3.9944 

3.9945 

3.9945 

3.9946 

3.9946 

3.9947 

3.9947 

44  40 

3.9948 

3.9948 

3.9948 

3.9949 

3.9949 

3.9950 

3.9950 

3.9951 

3.9951 

3.9952 

44  50 

3.9952 

3.9952 

3.9953 

3.9953 

3.9954 

39954 

3.9955 

3.9955 

3.9955 

3.9956 

2  45   0 

3.9956 

3.9957 

3.9957 

3.9958 

3.9958 

3.9959 

3.9959 

3.9959 

3.9960 

3.9960 

45  10 

3.9961 

3.9961 

3.9962 

3.9962 

3.9962 

3.9963 

3.9963 

3.9964 

3.9964 

3.9966 

45  20 

3.9965 

3.9966 

3.9966 

3.9966 

3.9967 

3.9967 

3.9968 

3.9968 

3.9969 

3.9969 

45  30 

3.9969 

3.9970 

3.9970 

3.9971 

3.9971 

3.9972 

3.9972 

3.9973 

3.9973 

3.9973 

45  40 

3.9974 

3.9974 

3.9975 

3.9975 

3.9976 

3.9976 

3.9976 

3.9977 

3.9977 

3.9978 

45  50 

3.9978 

3.9979 

3.9979 

3.9980 

3.9980 

3.9980 

3.9981 

3.9981 

3.9982 

3.9982 

2  46   0 

3.9983 

3.9983 

3.9983 

3.9984 

3.9984 

3.9985 

3.9985 

3.9986 

3.9986 

3.9987 

46  10 

3.9987 

3.9987 

3.9988 

3.9988 

3.9989 

3.9989 

3.9990 

3.9990 

3.9990 

3.9991 

46  20 

3.9991 

3.9992 

3,9992 

3.9993 

3.9993 

3.9993 

3.9994 

3.9994 

3.9995 

3.9995 

46  30 

3.9996 

3.9996 

3.9997 

3.9997 

3.9997 

3.9998 

3.9998 

3.9999 

3.9999 

4.0000 

46  40 

4.0000 

4.0000 

4.0001 

4.0001 

4.0002 

4.0002 

4.0003 

4.0003 

4.0003 

4.0004 

46  50 

4.0004 

4.0005 

4.0005 

4.0006 

4.0006 

4.0007 

4.0007 

4.0007 

4.0008 

4.0008 

2  47   0 

4.0009 

4.0009 

4.0010 

4.0010 

4.0010 

4.0011 

4.001  1 

4.0012 

4.0012 

4.0013 

47  10 

4.0013 

4.0013 

4.0014 

4.0014 

4.0015 

4.0015 

4.0016 

4.0016 

4.0016 

4.0017 

47  20 

4.0017 

4.0018 

4.0018 

4.0019 

4.0019 

4.0019 

4.0020 

4.0020 

4.0021 

4.0021 

47  30 

4.0022 

4.0022 

4.0023 

4.0023 

4.0023 

4.0024 

4.0024 

4.0025 

4.0025 

4.0026 

47  40 

4.0026 

4.0026 

4.0027 

4.0027 

4.0028 

4.002$ 

4.0029 

4.0029 

4.0029 

4.0030 

47  50 

4.0030 

4.0031 

4.0031 

4.0032 

4.0032 

4.0032 

4.0033 

4.0033 

4.0034 

4.0034 

2  48   0 

4.0035 

4.0035 

4.0035 

4.0036 

4.0036 

4.0037 

4.0037 

4.0038 

4.0038 

4.0038 

48  10 

4.0039 

4.0039 

4.0040 

4.0040 

4.0041 

4.0041 

4.0041 

4.0042 

4.0042 

4.0043 

48  20 

4.0043 

4.0044 

4.0044 

4.0045 

4.0045 

4.0045 

4.0046 

4.0046 

4.0047 

4.0047 

48  30 

4.0048 

4.0048 

4.0048 

4.0049 

4.0049 

4.0050 

4.0050 

4.0051 

4.0051 

4.0051 

48  40 

4.0052 

4.0052 

4.0053 

4.0053 

4.0054 

4.0054 

4.0054 

4.0055 

4.0055 

4.0056 

48  50 

4.0056 

4.0057 

4.0057 

4.0057 

4.0058 

4.0058 

4.0059 

4.0059 

4.0060 

4.0060 

2  49   0 

4.0060 

4.0061 

4.0061 

4.0062 

4.0062 

4.0063 

4.0063 

4.0063 

4.0064 

4.0064 

49  10 

4.0065 

4.0065 

4.0066 

4.0066 

4.0066 

4.0067 

4.0067 

4.0068 

4.0068 

4.0069 

49  20 

4.0069 

4.0069 

4.0070 

4.0070 

4.0071 

4.0071 

4.0072 

4.0072 

4.0072 

4.0073 

49  30 

4.0073 

4.0074 

4.0074 

4.0074 

4.0075 

4.0075 

4.0076 

4.0076 

4.0077 

4.0077 

49  40 

4.0077 

4.0078 

4.0078 

4.0079 

4.0079 

4.0080 

4.0080 

4.0080 

4.0081 

4.0081 

t9  50 

4.0082 

4.0082 

4.0083 

4.0083 

4.0083 

4.0084 

4.0084 

4.0085 

4.0085 

4.0086 

56 


TABLE    IX. 


LOGARITHMS  OF  SMALL  ARCS  IN  SPACE  OR  TIME. 

Arc. 

// 

0 

1 

2 

3 

• 

4 

5 

6 

n 

7 

8 

9 

0,    (    II 

2h.50m.  Qs. 

4.0086 

4.0086 

4.0087 

4.0087 

4.0088 

4.0088 

4.0089 

4.0089 

4.0089 

4.0090 

50  10 

4.0090 

4.0091 

4.0091 

4.0092 

4.0092 

4.0092 

4.0093 

4.0093 

4.0094 

4.0094 

50  20 

4.0095 

4.0095 

4.0095 

4.0096 

4.0096 

4.0097 

4.0097 

4.0097 

4.0098 

4.0098 

50  30 

4.0099 

4.0099 

4.0100 

4.0100 

4.0100 

4.0101 

4.0101 

4.0102 

4.0102 

4.0103 

50  40 

4.0103 

4.0103 

4.0104 

4.0104 

4.0105 

4.0105 

4.0106 

4.0106 

4.0106 

4.0107 

50  50 

4.0107 

4.0108 

4.0108 

4.0109 

4.0109 

4.0109 

4.0110 

4.0110 

4.0111 

4.0111 

2  51   0 

4.0111 

4.0112 

4.0112 

4.0113 

4.0113 

4.0114 

4.0114 

4.0114 

4.0115 

4.0115 

51  10 

4.0116 

4.0116 

4.0117 

4.0117 

4.0117 

4.0118 

4.0118 

4.0119 

40119 

4.0120 

51  20 

4.0120 

4.0120 

4.0121 

4.0121 

4.0122 

4.0122 

4.0122 

4.0123 

4.0123 

4.0124 

51  30 

4.0124 

4.0125 

4.0125 

4.0125 

4.0126 

4.0126 

4.0127 

4.0127 

4.0128 

4.0128 

51  40 

4.0128 

4.0129 

4.0129 

4.0130 

4.0130 

4.0130 

4.0131 

4.0131 

4.0132 

4.0132 

51  50 

4.0133 

4.0133 

4.0133 

4.0134 

4.0134 

4.0135 

4.01,  ']:,  '  4.0136 

4.0136 

4.0136 

2  52   0 

4.0137 

4.0137 

4.0138 

4.0138 

4.0138 

4.0139 

4.0!  I'.l 

4.0140 

4.0140 

4.0141 

52  10 

4.0141 

4.0141 

4.0142 

4.0142 

4.0143 

4.0143 

4.0.44 

4.0144 

4.0144 

4.0145 

52  20 

4.0145 

4.0146 

4.0146 

4.0146 

4.0147 

4.0147 

4.0148 

4.0148 

4.0149 

4.0149 

52  30 

4.0149 

4.0150 

4.0150 

4.0151 

4.0151 

4.0152 

4.0152 

4.0153 

4.0153 

4.0153 

52  40 

4.0154 

4.0154 

4.0154 

4.0155 

4.0155 

4.0156 

4.0156 

4.0157 

4.0157 

4.0157 

52  50 

4.0158 

4.0158 

4.0159 

4.0159 

4.0159 

4.0160 

4.0160 

4.0161 

4.0161 

4.0162 

2  53   0 

4.0162 

4.0162 

4.0163 

4.0163 

4.0164 

4.0164 

4.0164 

4.0165 

4.0165 

4.0166 

53  10 

4.0166 

4.0167 

4.0167 

4.0167 

4.0168 

•  4.0168 

4.0169 

4.0169 

4.0169 

4.0170 

53  20 

4.0170 

4.0171 

4.0171 

4.0172 

4.0172 

4.0172 

4.0173 

4.0173 

4.0174 

4.0174 

53  30 

4.0175 

4.0175 

4.0175 

4.0176 

4.0176 

4.0177 

4.0177 

4.0177 

4.0178 

4.0178 

53  40 

4.0179 

4.0179 

4.0180 

4.0180 

4.0180 

4.0181 

4.0181 

4.0182 

4.0182 

4.0182 

53  50 

4.0183 

4.0183 

4.0184 

4.0184 

4.0185 

4.0185 

4.0185 

4.0186 

4.0186 

4.0187 

2  54   0 

4.0187 

4.0187 

4.0188 

4.0188 

4.0189 

4.01  89 

4.0190 

4.0190 

4.0190 

4.0191 

54  10 

4.0191 

4.0192 

4.0192 

4.0192 

4.0193 

4.0193 

4.0194 

4.0194 

4.0194 

4.0195 

54  20 

4.0195 

4.0196 

4.0196 

4.0197 

4.0197 

4.0197 

4.0198 

4.0198 

4.0199 

4.0199 

54  30 

4.0199 

4.0200 

4.0200 

4.0201 

4.0201 

4.0202 

4.0202 

4.0202 

4.0203 

4.0203 

54  40 

4.0204 

4.0204 

4.0204 

4.0205 

4.0205 

4.0206 

4.0206 

4.0207 

4.0207 

4.0207 

54  50 

4.0208 

4.0208 

4.0209 

4.0209 

4.0209 

4.0210 

4.0210 

4.0211 

4.0211 

4.0211 

2  55   0 

4.0212 

4.0212 

4.0213 

4.0213 

4.0214 

4.0214 

4.0214 

4.0215 

4.0215 

4.0216 

55  10 

4.0216 

4.0216 

4.0217 

4.0217 

4.0218 

4.0218 

4.0219 

4.0219 

4.0219 

4.0220 

55  20 

4.0220 

4.0221 

4.0221 

4.0221 

4.0222 

4.0222 

4.0223 

4.0223 

4.0223 

4.0224 

55  30 

4.0224 

40225 

4.0225 

4.0225 

4.0226 

4.0226 

4.0227 

4.0227 

4.0228 

4.0228 

55  40 

4.0228 

4.0229 

4.0229 

4.0230 

4.0230 

4.0230 

4.0231 

4.0231 

4.0232 

4.0232 

55  50 

4.0233 

4.0233 

4.0233 

4.0234 

4.0234 

4.0235 

4.0235 

4.0235 

4.0236 

4.0236 

2  56   0 

4.0237 

4.0237 

4.0237 

4.0238 

4.0238 

4.0239 

4.0239 

4.0240 

4.0240 

4.0240 

56  10 

4.0241 

4.0241 

4.0242 

4.0242 

4.0242 

4.0243 

4.0243 

4.0244 

4.0244 

4.0244 

56  20 

4.0245 

4.0245 

4.0246 

4.0246 

4.0246 

4.0247 

4.0247 

4.0248 

4.0248 

4.0249 

56  30 

4.0249 

4.0249 

4.0250 

4.0250 

4.0251 

4.0251 

4.0251 

4.0252 

4.0252 

4.0253 

56  40 

4.0253 

4.0253 

4.0254 

4.0254 

4.0255 

4.0255 

4.0256 

4.0256 

4.0256 

4.0257 

56  50 

4.0257 

4.0258 

40258 

4.0258 

4.0259 

4.0259 

4.0260 

4.0260 

4.0260 

4.0261 

2  57   0 

4.0261 

4.0262 

4.0262 

4.0262 

4.0263 

4.0263 

4.0264 

4.0264 

4.0265 

4.0265 

57  10 

4.0265 

4.0266 

4.0266 

4.0267 

4.0267 

4.0267 

4.0268 

4.0268 

4.0269 

4.0269 

57  20 

4.0269 

4.0270 

4.0270 

4.0271 

4.0271 

4.0271 

4.0272 

4.0272 

4.0273 

4.0273 

57  30 

4.0273 

4.0274 

4.0274 

4.0275 

4.0275 

4.0276 

4.0276 

4.0276 

4.0277 

4.0277 

57  40 

4.0278 

4.0278 

4.0278 

4.0279 

4.0279 

4.0280 

4.0280 

4.0280 

4.0281 

4.0281 

57  50 

4.0282 

4.0282 

4.0282 

4.0283 

4.0283 

4.0284 

4.0284 

4.0284 

4.0285 

4.0285 

2  58   0 

4.02S6 

4.0286 

4.0287 

4.0287 

4.0287 

4.0288 

4.0288 

4.0289 

4.0289 

4.0289 

58  10 

4.0290 

4.0290 

4.0291 

4.0291 

4.0291 

4.0292 

4.0292 

4.0293 

4.0293 

4.0293 

58  20 

4.0294 

4.0294 

4.0295 

4.0295 

4.0295 

4.0296 

4.0296 

4.0297 

4.0297 

4.0297 

58  30 

4.0298 

4.0298 

4.0299 

4.0299 

4.0300 

4.0300 

4.0300 

4.0301 

4.0301 

4.0302 

58  40 

4.0302 

4.0302 

4.0303 

4.0303 

4.0304 

4.0304 

4.0304 

4.0305 

4.0305 

4.0306 

58  50 

4.0306 

4.0306 

4.0307 

4.0307 

4.0308 

4.0308 

4.0308 

4.0309 

4.0309 

4.0310 

2  59   0 

4.0310 

4.0310 

4.031  1 

4.0311 

4.0312 

4.0312 

4.0312 

4.0313 

4.0313 

4.0314 

59  10 

4.0314 

4.0314 

4.0315 

4.0315 

4.0316 

4.0316 

4.0317 

4.0317 

4.0317 

4.0318 

59  20 

4.0318 

4.0319 

4.0319 

4.0319 

4.0320 

4.0320 

4.0321 

4.0321 

4.0321 

4.0322 

59  30 

4.0322 

4.0323 

4.0323 

4.0323 

4.0324 

4.0324 

4.0325 

4.0325 

4.0325 

4.0326 

59  40 

4.0326 

4.0327 

4.0327 

4.0327 

4.0328 

4.0328 

4.0329 

4.0329 

4.0329 

4.0330 

59  50 

4.0330 

4.0331 

4.0331 

4.0331 

4.0332 

4.0332 

4.0333 

4.0333 

4.0333 

4.0334 

TABLE  X. 


TABLE    SHOWING    THE    CORRECTION    REQUIRED,  ON  ACCOUNT  OF 

SECOND  DIFFERENCES  OF  THE  MOON'S  MOTION,  IN  FINDING 

THE  GREENWICH  TIME   CORRESPONDING  TO  A 

CORRECTED  LUNAR  DISTANCE. 

Difference  of  the  Proportional  Logarithms  in  the  Ephemeris. 

Approximate 

I 

4 

6 

8 

10 

12  14  16 

IS 

20 

2224 

26 

28 

§032 

34 

36 

38 

40 

42 

44  46  48 

ftjfl 

hn 

. 

0     0 

3     0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

C 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

C 

0 

0  10 

2  50 

0 

0 

0 

1 

i 

1 

1 

1 

i 

i 

1 

2 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

8 

3 

3 

0  20 

2  40 

0 

1 

1 

1 

i 

2 

2 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

4 

5 

5 

5 

5 

6 

6 

6 

6 

0  30 

2  30 

0 

1 

2 

2 

2 

2 

3 

3 

3 

4 

4 

5 

5 

5 

C 

G 

G 

7 

7 

7 

8 

8 

8 

9 

9 

0  40 

2  20 

0 

1 

2 

2 

3 

3 

3 

4 

4 

5 

5 

6 

6 

G 

7 

7 

8 

8 

9 

9 

10 

10 

10 

11 

11 

0  50 

2  10 

1 

2 

2 

3 

3 

4 

4 

5 

5 

5 

G 

6 

7 

7 

8 

8 

9 

9 

10 

10 

11 

12 

12 

13 

13 

1     0 

2     0 

1 

2 

2 

3 

3 

4 

4 

B 

C 

6 

7 

7 

8 

8 

9 

9 

10 

10 

11 

12 

12 

13 

13 

14 

14 

1  10 

1  50 

I 

2 

2    3 

4 

4    5 

5 

G 

C 

7 

8 

8 

9 

9 

10 

11 

11 

12 

12 

13 

14 

14 

15 

15 

1  20 

1  40 

I 

2 

3    3 

4 

4    5 

C 

G 

7 

7 

8 

9 

9  10 

10 

11 

12 

12 

13 

14 

14 

15 

15 

16 

1  30 

1  30 

1 

2 

3 

3 

4 

4 

5 

G 

G 

7 

8 

8 

9 

9  10 

11 

11 

12 

12 

13 

14 

14 

15 

If 

16 

Difference  of  the  Proportional  logarithms  in  the  Ephemeris, 

§4 

5553 

63 

63  64  65 

G3 

70 

72 

74 

76  78  80 

82 

8486 

S3 

9092 

91 

9698 

100 

102 

. 

h.  ra 

0     0 

Olft 

3     0 
2r:n 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

7! 

0  20 

2  40 

7 

7 

7 

1 

8 

8 

8 

8 

9 

9 

9 

9 

10 

10 

10 

10 

11 

11 

11 

11 

12 

12 

12 

12 

13 

0  30 

2  30 

9 

10 

10 

10 

11 

11 

12 

12 

12 

13 

13 

13 

14 

14 

14 

14 

15 

15 

1G 

16 

16 

17 

17 

17 

18 

0  40 

2  20 

12 

12 

13 

13 

13  14  14 

15  15  16 

1G 

1C 

17 

17 

18 

18 

19 

19  19 

20 

20 

21 

21 

22 

22 

0  50 

2  10 

14 

14 

15 

15 

16  16 

16 

17 

17 

IS 

19 

19 

20 

20 

21 

21 

22 

22  22 

23 

23 

24 

24 

25 

26 

1     0 

2     0 

15 

16 

16 

17 

17 

18 

18 

19 

19 

20 

21 

21 

22 

22 

23  23 

24 

24 

25 

25 

26 

27 

27 

28 

28 

1  10 

1  50 

10 

17 

17 

18  18  19  19  2021  21 

22  22  23 

24  24  25  25 

2G 

27 

27 

28 

28 

29 

30 

30 

1  20 

1  40 

17 

17 

18 

19  19  20  20  21 

21  22 

23  23  24  25  25  26  26 

27 

28 

28 

29 

29 

30 

31 

31 

1  30 

1  30 

17 

18 

18 

19 

19  20  21  21 

22  23 

23  24  24  25  25  26  27 

27 

28 

29 

29 

30  31 

31 

32 

1      1 

1 

Difference  of  the  Proportional  Logarithms  in  the  Ephemeris, 

W. 

I  106 

1(E 

110 

112 

114 

115 

118 

120 

122   124 

i: 

i  128 

132 

132 

131 

136 

188 

. 

. 

0     0 

3     0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0       0 

0 

0 

0 

0 

0 

0 

0 

0  10 

2  50 

7 

7 

7 

7 

7 

7 

8 

8 

8 

8       8 

8 

8 

8 

9 

9 

9 

9 

0  20 

2  40 

13 

13 

13 

14 

14 

14 

14 

15 

15 

15     15 

15 

16 

10 

16 

1G 

17 

17 

0  30 

2  30 

18 

13 

19 

19 

19 

20 

20 

20 

21 

21     21 

22 

22 

22 

23 

23 

24 

24 

0  40 

2  20 

22 

23 

23 

24 

24 

25 

25 

25 

26 

25     27 

27 

23 

23 

23 

29 

29 

30 

0  50 

2  10 

26 

26 

27 

27 

28 

29 

£9 

29 

30 

30     31 

31 

32 

32 

33 

33 

34 

34 

1     0 

2     0 

29 

29 

30 

30 

31 

31 

32 

33 

33 

34     34 

35 

35 

3G 

37 

37 

38 

38 

1   10 

1   50 

31 

31 

32 

32 

33 

34 

34 

35 

35 

36     37 

37 

33 

33 

39 

40 

40 

41 

1  20 

1  40 

32 

33 

33 

34 

34 

35 

35 

36 

37 

33     33 

39 

39 

43 

41 

41 

42 

42 

1  30 

1  30 

32 

33 

34 

34 

35 

35 

3G 

36 

37 

33     39 

39 

40 

40 

41 

42 

42 

43 

The  Correction  is  to  be  aslded  to  the  approximate  Greenwich  Time  when  the  Proportional  Logarithms  in  the 

Epheineris  are  decreasing^  and  subtracted  when  they  are  increasing. 

58 


TABLE    XI. 


LOG.  N  FOR  DISTANCES  FROM  THE  SUN. 

ISff. 

Qh. 

f" 

6h- 

gh. 

ir 

15h 

18h 

21h. 

18ff. 

oh- 

r 

6h- 

9b- 

ir 

IS"' 

18h- 

21h. 

Jan.    7 

—0.76 

.78 

.79 

.80 

.81 

.82 

.83 

.84 

Apr.  10 

+0.66 

.69 

.72 

.74 

.77 

.79 

.81 

.83 

8 

0.85 

.86 

.87 

.87 

.88 

.89 

.89 

.90 

11 

0.85 

0.86 

088 

0.89 

0.91 

0.92 

0.93 

0.95 

9 

0.90 

.91 

.91 

.91 

.92 

.92 

.92 

.92 

12 

0.96 

0.97 

0.98 

0.99 

1.00 

1.01 

1.02 

1.02 

10 

0.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

13 

+1.03 

1.04 

1.04 

1.05 

1.06 

1.06 

1.07 

1.07 

11 

0.92 

.91 

.91 

.91 

.90 

.90 

.89 

.89 

18 

—0.80 

0.78 

0.77 

0.76 

0.75 

0.73 

0.71 

0.69 

12 

—0.88 

.87 

.86 

.85 

.84 

.83 

.82 

.81 

19 

—0.67 

.65 

.63 

.60 

.58 

.55 

.52 

.48 

13 

0.80 

.78 

.77 

.75 

.73 

.71 

.69 

.66 

20 

—0.46 

.41 

.36 

.31 

.25 

.18 

.10 

.00 

14 

0.63 

.60 

.56 

.52 

.48 

.42 

.36 

.28 

22 

+0.12 

.20 

.26 

.32 

.37 

.41 

.45 

.49 

20 

0.71 

.74 

.76 

.79 

.81 

.83 

.84 

.85 

23 

0.52 

.55 

.58 

.60 

.63 

.65 

.67 

.69 

21 

0.87 

.88 

.89 

.89 

.90 

.91 

.91 

.92 

24 

0.71 

.73 

.75 

.76 

.78 

.79 

.80 

.82 

22 

—0.92 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

25 

+0.83 

.84 

.85 

.86 

.87 

.88 

.89 

.90 

23 

0.93 

.93 

.93 

.92 

.92 

.91 

.91 

.90 

26 

0.91 

.92 

.93 

.93 

.94 

.95 

0.95 

0.96 

24 

0.90 

.89 

.89 

.88 

.87 

.86 

.85 

.84 

27 

0.97 

0.97 

0.98 

.98 

.99 

.99 

0.99 

1.00 

25 

0.83 

.82 

.81 

.80 

.78 

.77 

.75 

.74 

May  6 

9.77 

9.96 

0.10 

.20 

.28 

.35 

0.40 

0.46 

26 

0.72 

.70 

.68 

.66 

.04 

.61 

.59 

.56 

7 

0.50 

0.54 

0.58 

.61 

.04 

.67 

.70 

.72 

27 

—0.53 

.49 

.45 

.41 

.37 

.31 

.25 

.18 

8 

+0.75 

.77 

.79 

.80 

.82 

.84 

.85 

.87 

Feb.    6 

0.88 

.88 

.88 

.89 

.89 

.89 

.89 

.89 

9 

0.88 

.89 

.90 

.92 

.93 

.94 

0.94 

0.95 

7 

0.89 

.89 

.89 

.89 

.89 

.88 

.88 

.88 

10 

0.96 

0.97 

0.97 

0.98 

0.99 

0.99 

0.99 

LOO 

8 

0.87 

.87 

.86 

.86 

.85 

.84 

.83 

.82 

11 

1.00 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

101 

9 

0.82 

.80 

.79 

.78 

.77 

.75 

.74 

.72 

12 

1.01 

1.01 

1.01 

1.01 

1.01 

1.00 

1.00 

1.00 

10 

—0.70 

.68 

.66 

.64 

.61 

.58 

.55 

.52 

13 

+0.99 

0.09 

0.98 

0.97 

0.97 

0.96 

0.95 

0.94 

11 

—0.48 

.43 

.38 

.32 

.25 

.16 

.05 

9.89 

18 

—0.46 

.41 

.35 

.29 

.22 

.13 

0.03 

9.89 

13 

+0.43 

.49 

.55 

.60 

.64 

.69 

.73 

.76 

20 

+0.23 

.29 

.34 

.39 

.43 

.47 

0.51 

0.54 

14 

+0.80 

.83 

.87 

.90 

.93 

.96 

.99 

1.02 

21 

0.57 

.59 

.62 

.64 

.66 

.68 

.70 

.72 

18 

—0.78 

.80 

.82 

.84 

.85 

.86 

.87 

.88 

22 

0.73 

.75 

.76 

.78 

.79 

.81 

.82 

.83 

19 

—0.89 

.89 

.90 

.90 

.90 

.90 

.90 

.90 

23 

+0.84 

.85 

.86 

.87 

.88 

.88 

.89 

.90 

20 

0.90 

.89 

.89 

.89 

.88 

.88 

.87 

.86 

24 

0.91 

.91 

.92 

.93 

.93 

.94 

.94 

.94 

21 

0.86 

.85 

.84 

.83 

.82 

.81 

.79 

.78 

25 

0.95 

.95 

.96 

.96 

.96 

.96 

.96 

.97 

22 

0.77 

.75 

.74 

.72 

.70 

.69 

.67 

.64 

26 

0.97 

.97 

.97 

.97 

.97 

.97 

.97 

.96 

23 

0.62 

.60 

.57 

.54 

.51 

.48 

.44 

.40 

27 

0.96 

.96 

.96 

.95 

.95 

.94 

.94 

.98 

26 

+0.30 

.36 

.41 

.45 

.49 

.53 

.56 

.60 

June  3 

+0.05 

.18 

.28 

.35 

.42 

.47 

.52 

.56 

Mar.  8 

—0.83 

.82 

.81 

.80 

.79 

.78 

.77 

.76 

4 

0.60 

.63 

.66 

.69 

.72 

.74 

.76 

.78 

9 

0.74 

.73 

.71 

.70 

.68 

.66 

.64 

.60 

5 

0.80 

.82 

.83 

.85 

.86 

.87 

.88 

.89 

10 

—0.59 

.56 

.53 

.49 

.45 

.41 

.36 

.30 

6 

0.90 

.91 

.92 

.93 

.94 

.94 

.95 

.95 

12 

+0.91 

.06 

.17 

.26 

.34 

.40 

.46 

.51 

7 

0.96 

.96 

.97 

.97 

.97 

.97 

.98 

.98 

13 

+0.55 

.59 

.63 

.67 

.70 

.73 

.76 

.78 

8 

+0.98 

.98 

.98 

.98 

.97 

.97 

.97 

.97 

14 

0.81 

0.83 

0.85 

0.87 

0.89 

0.91 

0.93 

0.95 

9 

0.96 

.96 

.95 

.95 

.94 

.94 

.93 

.92 

15 

+0.97 

0.99 

1.00 

1.02 

1.03 

1.05 

1.06 

1.08 

10 

0.91 

.91 

.90 

.88 

.87 

.86 

.85 

.83 

19 

—0.78 

0.80 

0.82 

0.84 

0.84 

0.85 

085 

0.86  !          11 

0.82 

.80 

.78 

.76 

.73 

.71 

.68 

.64 

20 

—0.86 

.86 

.85 

.85 

.84 

.84 

.83 

«            17 

0.18 

.26 

.32 

.38 

.43 

.47 

.51 

.54 

21 

—0.82 

.81 

.80 

.78 

.77 

.76 

.74 

.73 

18 

+0.57 

.60 

.63 

.65 

.67 

.69 

.71 

.73 

22 

0.71 

.69 

.67 

.65 

.63 

.61 

.58 

.55 

19 

0.74    .76!   .77 

.79 

.80 

.81 

.82 

.83 

23 

—0.53 

.49 

.46 

.42 

.38 

.33 

.27 

.21 

20 

0.84  '   .85     .86 

.87 

.87 

.88 

.89 

.89 

25 

+0.84 

.98 

.08 

.16 

.23 

.29 

.35 

.39 

21 

0.90 

.90 

.91 

.91 

.92 

.92 

.93 

.93 

26 

0.44 

.48 

.51 

.54 

.57 

.60 

.63 

.65 

22 

0.93 

.93 

.92 

.94 

.94 

.94 

.94 

.94 

27 

+0.67 

.69 

.71 

.73 

75 

.77 

.79 

.80 

23 

+0.94 

.94 

.94 

.93 

.93 

.93 

.93 

.92 

28 

+0.82 

.83 

.85 

.86 

.87 

.88    .90 

.91 

24 

0.92 

.91 

.91 

.90 

.90 

.89 

.88 

.88 

Apr.    6 

—0.67 

.65 

.63 

.61 

.58 

.55'    .52 

.48 

25 

0.87 

.86 

.85     .83 

.82 

.81 

.79 

.78 

7 

-  0.44 

.40 

.35 

.29 

.22'   .13'   .03 

.88 

Julv    3 

0.82    .84 

.85     .87 

.88 

.89 

.90 

.91 

9 

+0.31 

.37 

.43 

.48 

.52     .56     .60 

.63 

'    4 

0.92 

.92 

.93     .93 

.93 

.94 

.95 

.95 

59 


TABLE    XI. 


LOG.  N  FOR  DISTANCES  FROM  THE  SUN. 

184$. 

Qh. 

r 

gh. 

gh. 

ir 

IF' 

18h- 

21h- 

18tf. 

Qh. 

8h- 

gh. 

Qh. 

ir 

15h- 

18h> 

21m 

July   5 

+0.95 

.95 

.96 

.96 

.96 

.95 

.95 

.95 

Oct.    3 

—0.40 

.44 

.48 

.52 

.55 

.58 

.61 

.63 

6 

0.95 

.95 

.94 

.94 

.94 

.93 

.93 

.92 

4 

0.66 

.68 

.70 

.72 

.74 

.76 

.77 

.79 

7 

0.91 

.91 

.90 

.89 

.88 

.87 

.86 

.85 

5 

0.80 

.82 

.83 

.84 

.86 

.87 

.88 

.89 

8 

0.84 

.83 

.81 

.80 

.78 

.76 

.75 

.73 

6 

0.90 

.91 

.92 

0.93 

0.94 

0.95 

0.96 

0.96 

9 

0.70 

.68 

.66 

.63 

.60 

0.57 

0.53 

0.49 

7 

—0.97 

.98 

.98 

0.99 

1.00 

1.00 

1.01 

1.02 

10 

+0.44 

.39 

.38 

.25 

.16 

0.04 

9.86 

9.54 

13 

+0.84 

.83 

.82 

0.81 

0.80 

0.78 

0.77 

0.76 

16 

0.47 

.52 

.56 

.60 

.63 

0.66 

0.68 

0.70 

14 

0.74 

.72 

.71 

.69 

.67 

.64 

.62 

.59 

17 

0.72 

.74 

.76 

.77 

.79 

.80 

.81 

.82 

15 

+0.56 

.53 

.50 

.46 

.42 

.37 

.31 

.25 

18 

0.83 

.84 

.85 

.86 

.86 

.87 

.88 

.88 

17 

—9.98 

.10 

.20 

.28 

.34 

.40 

.45 

.50 

19 

0.89 

.89 

.90 

.90 

.90 

.91 

.91 

.91 

18 

0.54 

.58 

.61 

.64 

.67 

.70 

.73 

.75 

20 

+0.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

19 

—0.77 

.79 

.81 

.83 

.85 

.86 

.88 

.89 

21 

0.91 

.91 

.90 

.90 

.90 

.89 

.89 

.88 

20 

0.91 

.92 

.93 

.94 

.96 

.97 

.98 

.99 

22 

0.88 

.87 

.86 

.85 

.85 

.84 

.83 

.82 

30 

9.99 

.10 

.18 

.25 

.31 

.36 

.41 

.45 

23 

0.80 

.79 

.78 

.76 

.75 

.73 

.71 

.69 

31 

0.49 

.52 

.55 

.58 

.61 

.63 

.65 

.68 

24 

0.67 

.64 

.62 

.59 

.55 

.52 

.47 

.42 

Nor.  1 

0.70 

.72 

.73 

.75 

.77 

.78 

.79 

.81 

Aug.  1 

+0.91 

.92 

.92 

.93 

.93 

.93 

.94 

.94 

2 

—0.82 

.83 

.84 

.85 

.86 

.87 

.88 

.89 

2 

0.94 

.94 

.94 

.94 

.93 

.93 

.93 

.92 

3 

0.90 

.91 

.91 

.92 

.93 

.93 

.94 

.94 

3 

0.92 

.91 

.91 

.90 

.90 

.89 

.88 

.87 

4 

0.95 

.95 

.96 

.96 

.96 

.97 

.97 

.97 

4 

0.86 

.85 

.86 

.83 

.82 

.80 

.79 

.77 

5 

0.98 

.98 

.98 

.98 

.98 

.98 

.98 

.98 

5 

0.76 

.74 

.72 

.70 

.68 

.66 

.63 

.61 

6 

—0.98 

.98 

.98 

.98 

.98 

.97 

.97 

.97 

6 

+0.58 

.55 

.51 

.47 

.43 

.38 

.83 

.27 

12 

+0.53 

.48 

.43 

.37 

.31 

.23 

.14 

.02 

8 

—9.93 

.07 

.17 

.26 

.33 

.39 

.45 

.50 

14 

—0.28 

.35 

.41 

.46 

.50 

.54 

.58 

.61 

9 

—0.54 

.58 

.62 

.66 

.69 

.72 

.75 

.78 

15 

0.64 

.67 

.76 

.72 

.74 

.77 

.79 

.80 

15 

+0.79 

.81 

.82 

.83 

.84 

.85 

.86 

.86 

16 

0.82 

.84 

.85 

.87 

.88 

.89 

.90 

.91 

16 

0.87 

.87 

.88 

.88 

.88 

.89 

.89 

.89 

17 

0.92 

.93 

.94 

.95 

.96 

0.96 

0.97 

0.97 

17 

+0.89 

.89 

.89 

.89 

.89 

.88 

.88 

.88 

18 

—0.98 

0.98 

0.99 

0.99 

0.99 

1.00 

1.00 

1.00 

18 

0.88 

.87 

.87 

.86 

.86 

.85 

.84 

.84 

19 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

0.99 

0.99 

19 

0.83 

.82 

.81 

.80 

.79 

.78 

.76 

.75 

27 

0.06 

0.16 

0.24 

0.31 

0.36 

0.41 

.45 

.49 

20 

0.73 

.72 

.70 

.68 

.66 

.64 

.61 

.58 

28 

0.53 

.56 

.59 

.62 

.64 

.66 

.68 

.70 

21 

+0.55 

.52 

.48 

.44 

.40 

.34 

.28 

.20 

29 

0.72 

.73 

.75 

.76 

.78 

.79 

.80 

.82 

23 

—0.17 

.27 

.35 

.42 

.48 

.53 

.58 

.63 

30 

—0.83 

.84 

.85 

.86 

.86 

.87 

.88 

.89 

30 

+0.92 

.92 

.92 

.91 

.91 

.91 

.90 

.90 

Dec.    1 

0.89 

.90 

.91 

.91 

.92 

.92 

.93 

.93 

31 

0.89 

.88 

.87 

.87 

.86 

.85 

.84 

.82 

2 

0.93 

.94 

.94 

.94 

.94 

.95 

.95 

.95 

Sept.  1 

0.81 

.70 

.78 

.77 

.75 

.74 

.72 

.70 

3 

0.95 

.95 

.95 

.95 

.95 

.95 

.94 

.94 

2 

0.67 

.65 

.63 

.60 

.57 

.54 

.51 

.47 

4 

0.94 

.94 

.93 

.93 

.93 

.92 

.92 

.91 

3 

+0.43 

.38 

.33 

.27 

.20 

.12 

.02 

.88 

5 

—0.90 

.90 

.89 

,88 

.87 

.86 

.85 

.84 

5 

—0.24 

.31 

.36 

.41 

.46 

.50 

.53 

.57 

6 

0.82 

.81 

.79 

.77 

.75 

.73 

.70 

.67 

6 

0.60 

.63 

.65 

.68 

.70 

.73 

.75 

.77 

12 

0.38 

.44 

.49 

.54 

.58 

.62 

.65 

.68 

7 

—0.79 

.81 

.83 

.84 

.86 

.88 

.89 

.71 

13 

0.71 

.73 

.75 

.77 

.79 

.81 

.83 

.84 

13 

+0.83 

.84 

.85 

.85 

.86 

.86 

.86 

.87 

14 

0.86 

.87 

.88 

.89 

.90 

.91 

.92 

.93 

14 

+0.87 

.87 

.87 

.86 

.86 

.86 

.86 

.85 

15 

—0.93 

.94 

.95 

.95 

.96 

.96 

.96 

.97 

15 

0.85 

.84 

.83 

.83 

.82 

.81 

.80 

.79 

16 

0.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

16 

0.78 

.77 

.75 

.74 

.72 

.71 

.69 

.67 

17 

0.97 

.96 

.96 

.96 

.95 

.95 

.94 

.94 

17 

0.65 

.63 

.60 

.58 

.55 

0.52 

0.48 

0.44 

18 

0.93 

.92 

.91 

.90 

.89 

.88 

.87 

.86 

18 

+0.40 

.35 

.29 

.22 

.14 

0.03 

9.89 

9.67 

27 

0.72 

.73 

.75 

.76 

.78 

.79 

.80 

.81 

20 

—0.39 

.44 

.49 

.54 

.58 

0.62 

0.65 

0.69 

28 

—0.83 

.84 

.84 

.85 

.86 

.87 

.88 

.88 

21 

-0.71 

.74 

.77 

.79 

.82 

.84 

.86 

.88 

29 

0.89 

.89 

.90 

.90 

.91 

.91 

.91 

.92 

29 

+0.77 

.75 

.74 

.72 

.70 

.68 

.65 

.63 

30 

0.92 

.92 

.92 

.92 

.93 

.93 

.93 

.93 

30 

+0.60 

0.57 

0.54 

0.50 

.46 

.42 

.37 

.32 

31 

—0.93 

.92 

.92 

.92 

.92 

.92 

.91 

.91 

Oct.    2 

—9.58 

9.81 

9.97 

0.08 

.16 

.24 

.30 

.35 

60 


TABLE    XII. 

For  finding  the  valtre  of  N  for  correcting  lunar  distances  for  the  compression  of  the  earth. 


TABLE  XII.  A.  giving  1st  Part  of  N. 

TABLE  XII.  B.  giving  2d  Part  of  N. 

ADD 

J)'s  Declination. 

App. 

j|C's  Declination. 

A^JJ. 

Dist. 

0 

3° 

o 

1 

12 

0 

15 

ft 

il 

0 

24 

27 

30 

Dist. 

o 

I 

6 

o 

9 

12 

o 

15 

1°8 

21 

& 

c 

27 

30 

0 

20 

o 
20 

4-0 

i 

II 

1 

a 

10 

// 

14 

i? 

26' 

// 

24 

27 

// 

30 

II 

33 

—  0 

Si    6 

10 

13 

16 

19 

22 

25 

28 

31 

22 

0 

3 

6 

9 

12 

14 

17 

20 

23 

25 

28 

22 

0 

3 

6 

9 

13 

16 

19 

22 

29 

•27 

30 

24 

0 

3 

5 

8 

11 

13 

16 

18 

21 

23 

25 

24 

0 

a 

6 

9 

12 

14 

17 

20 

23 

25 

28 

26 

0 

2 

5 

7 

10 

12 

14 

17 

19 

21 

23 

26 

0 

3 

5 

8 

11 

13 

16 

18 

21    23 

26 

28 

0 

2 

4 

7 

9 

11 

13 

15    17 

19 

21 

28 

0 

3 

5 

8 

1C) 

12 

15 

17 

20 

22 

24 

30 

—  0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

30 

4-0 

2 

5 

7 

9 

12 

14 

16 

18 

21 

23 

32 

0 

2 

4 

6 

8 

9 

11 

13 

15 

16 

18 

32 

0 

2 

4 

7 

9 

11 

13 

15 

17 

19 

21 

34 

0 

2 

4 

5 

7 

y 

10 

12 

14 

15 

17 

34 

0 

2 

4 

6 

8 

11 

13 

15 

10 

18 

20 

36 

0 

2 

3 

5 

7 

8 

10 

11 

13 

14 

16 

36 

0 

2 

4 

6 

8 

10 

12 

14 

10 

17 

19 

38 

0 

2 

3 

0 

6 

8 

9 

10 

12 

13 

14 

38 

0 

2 

4 

6 

8 

10 

11 

13 

15 

17 

18  ; 

40 

—  0 

1 

3 

4 

6 

7 

8 

10 

11 

12 

13 

40 

4-0 

2 

4 

6 

7 

9 

11 

13 

14 

16 

18 

42 

0 

1 

3 

4 

5 

7 

8 

9 

10 

11 

13 

42 

0 

2 

4 

5 

7 

9 

10 

12 

14 

is 

17  , 

44 

0 

1 

2 

4 

5 

6 

7 

8 

10 

11 

12 

44 

0 

2 

3 

5 

7 

8 

10 

12 

13 

15 

16 

46 

0 

1 

2 

3 

5 

6 

7 

8 

9 

10 

11 

46 

0 

2 

3 

5 

6 

8 

10 

11 

ia 

14 

16 

48 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

48 

0 

2 

3 

5 

6 

8 

9 

11 

12 

14 

15 

50 

—  0 

2 

a 

4 

5 

6 

7 

8 

9 

10 

50 

4-0 

2 

3 

5 

6 

8 

9 

11 

12 

ia 

15  • 

52 

0 

2 

3 

4 

5 

5 

6 

7 

8 

9 

52 

0 

2 

3 

4 

6 

7 

9 

10 

12 

13 

14  i 

54 

0 

2 

3 

3 

4 

5 

6 

7 

7 

8 

54 

0 

1 

3 

4 

6 

7 

9 

10 

11 

13 

14  | 

56 

0 

2 

2 

3 

4 

5 

5 

6 

7 

8 

56 

0 

1 

3 

4 

6 

7 

8 

10 

11 

i  a 

14  | 

58 

0 

1 

2 

3 

4 

4 

5 

6 

6 

7 

58 

0 

1 

3 

6 

7 

8 

10 

11 

H 

13  , 

60 

—  0 

1 

2 

3 

3 

4 

5 

5 

6 

7 

60 

4-0 

3 

5 

7 

8 

9 

11 

12 

13  !' 

62 

0 

1 

2 

3 

3 

4 

4 

5 

5 

6 

62 

0 

3 

5 

7 

8 

9 

10 

\-> 

13  ; 

64 

0 

1 

2 

2 

3 

3 

4 

4 

5 

6 

64 

0 

3 

5 

7 

8 

9 

10 

11 

13 

66 

0 

1 

2 

2 

3 

3 

4 

4 

5 

5 

66 

0 

3 

5 

6 

8 

9 

10 

11 

12  : 

68 

0 

0 

1 

2 

2 

3 

3 

4 

4 

5 

68 

0 

3 

5 

6 

8 

9 

10 

11 

12  ; 

70 

—  0 

0 

1 

2 

2 

3 

3 

3 

4 

4 

70 

4-0 

3 

4 

B 

6 

7 

9 

10 

11 

12  ! 

72 

0 

0 

] 

2 

2 

2 

3 

3 

3 

4 

72 

0 

2 

4 

5 

6 

7 

9 

10 

11 

12  i 

74 

0 

0 

1 

1 

2 

2 

2 

3 

3 

3 

74 

0 

2 

4 

5 

6 

7 

8 

10 

11 

12 

76 

0 

0 

1 

1 

1 

2 

2 

2 

3 

3 

76 

0 

2 

4 

5 

6 

7 

8 

9 

11 

12 

78 

0 

0 

0 

1 

1 

1 

1 

2 

2 

2 

2 

78 

0 

2 

4 

5 

6 

7 

8 

9 

11 

12 

80 

—  0 

0 

0 

1 

1 

1 

1 

1 

2 

2 

2 

80 

4-0 

2 

4 

5 

6 

7 

8 

9 

10 

11   i 

82 

0 

0 

0 

0 

1 

1 

1 

1 

1 

] 

2 

82 

0 

2 

4 

5 

6 

7 

8 

9 

10 

11 

84 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

84 

0 

2 

4 

5 

6 

7 

8 

9 

10 

11 

86 

0 

0 

0 

0 

0 

0 

0 

1 

1 

1 

1 

86 

0 

2 

4 

5 

6 

7 

8 

9 

10 

11 

88 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

88 

0 

2 

4 

5 

6 

7 

8 

9 

10 

11 

90 

—  0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

90 

4-0 

2 

4 

5 

6 

7 

8 

9 

10 

11 

92 

+0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

92 

0 

2 

4 

5 

6 

7 

8 

9 

10 

ll  ; 

94 

0 

0 

0 

0 

0 

0 

0 

1 

1 

1 

1 

94 

0 

2 

4 

5 

6 

7 

8 

9 

10 

11  i 

96 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

96 

0 

2 

4 

5 

6 

7 

8 

9 

10 

11  1 

98 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

2 

98 

0 

2 

4 

5 

6 

7 

8 

9 

10 

11 

100 

+0 

0 

0 

1 

1 

1 

1 

2 

2 

2 

100 

4-0 

2 

4 

5 

6 

7 

8 

9 

10 

11 

102 

0 

0 

0 

1 

1 

1 

2 

2 

2 

2 

102 

0 

2 

4 

5 

6 

7 

8 

9 

11 

12 

104 

0 

0 

1 

I 

2 

2 

2 

3 

3 

104 

0 

2 

4 

5 

6 

7 

8 

9 

11 

12 

106 

0 

0 

1 

2 

2 

2 

3 

3 

3 

106 

0 

2 

4 

5 

6 

7 

8 

]() 

!1 

12  ' 

108 

0 

0 

2 

2 

2 

3 

3 

3 

4 

108 

0 

2 

4 

5 

6 

7 

9 

10 

11 

12 

110 

+0 

0 

1 

2 

2 

3 

3 

3 

4 

4 

110 

4-0 

3 

4 

5 

6 

7 

9 

10 

11 

12 

112 

0 

0 

1 

2 

2 

3 

3 

4 

4 

5 

112 

0 

3 

4 

5 

6 

8 

9 

10 

11 

12 

114 

0 

1 

2 

2 

3 

3 

4 

4 

5 

5 

114 

0 

3 

4 

5 

6 

8 

9 

10 

11 

12 

116 

0 

1 

2 

2 

3 

3 

4 

4 

5 

6 

116 

0 

3 

4 

5 

7 

8 

9 

10 

11 

13 

118 

0 

1 

2 

3 

3 

4 

4 

5 

5 

6 

118 

0 

3 

4 

5 

7 

8 

9 

10 

12 

13 

120 

+0 

1 

1 

2 

3 

3 

4 

5 

5 

6 

7 

120 

4-0 

1 

3 

4 

5 

7 

8 

9 

11 

12 

13 

122 

0 

1 

1 

2 

3 

4 

4 

5 

6 

6 

7 

122 

0 

1 

3 

4 

6 

7 

8 

10 

11 

12 

13 

124 

0 

1 

2 

2 

3 

4 

5 

5 

6 

7 

8 

124 

0 

1 

3 

4 

6 

7 

8 

10 

11 

12 

14 

126 

0 

1 

2 

a 

3 

4 

5 

6 

7 

7 

8 

126 

0 

1 

3 

4 

6 

7 

9 

10 

11 

i  a 

14 

128 

0 

1 

2 

3 

4 

5 

5 

6 

7 

8 

9 

128 

0 

2 

3 

4 

6 

7 

9 

10 

12 

13 

14 

130 

+0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

130 

4-0 

2 

3 

5 

6 

8 

9 

11 

12 

13 

15 

The  signs  in  the  0°  column  apply  to  all  the  numbers  in  the  same  line,  and  are  to  be  used  when  the  declination 
is  North.    When  the  declination  is  South,,  change  the  sign  +  to  —  and  —  to  +. 

61 


IMPROVED    METHOD 


OF  FINDING  THE 


3RROR  AND  HATE  OF  A  CHRONOMETER 
BY  EQUAL       ALTITUDES. 


FROM  THE  APPENDIX  TO  THE  AMERICAN  EPHEMERIS  AND  NAUTICAL 
ALMANAC  FOR  1866. 


METHOD 


OF   FINDING   THE 


ERROR  AND  RATE  OF  A  CHRONOMETER  BY  EQUAL  ALTITUDES. 


To  regulate  a  chronometer  to  Greenwich  time,  we  must  determine  its  error  and 
rate  at  a  place  whose  longitude  is  well  known.  The  most  accurate  method  of  doing 
this  is  by  observing  the  transit  of  the  sun  or  a  star  over  the  meridian.  For  the  nav- 
igator, the  most  simple  and  accurate  substitute  for  the  meridian  observation  is  that  cf 
equal  altitudes  of  the  same  object  on  each  side  of  the  meridian.  In  the  case  of  a 
star,  the  mean  of  the  two  chronometer  times  corresponding  to  the  equal  altitudes  is 
the  chronometer  time  of  transit ;  but  in  the  case  of  the  sun,  the  mean  of  these  times 
differs  somewhat  from  the  time  of  transit,  since,  in  consequence  of  the  change  of  the 
sun's  declination  between  the  observations,  the  equal  altitudes  do  not  occur  at  equal 
intervals  before  and  after  the  transit. 

The  small  correction  necessary,  when  the  sun  is  observed,  to  reduce  the  mean  of 
the  times  to  the  time  of  transit,  is  called  the  Equation  of  Equal  Altitudes. 

The  method  of  computing  this  equation  given  below  is  based  upon  that  first  given 
by  GAUSS  (Monatliche  Correspondenz,  Vol.  XXIII. ).  We  do  not,  however,  follow 
him  in  using  the  doable  daily  change  of  declination,  or  difference  between  the  sun's 
declination  on  the  noon  preceding  and  the  noon  following  that  of  the  observation ;  but 
prefer  to  use  the  hourly  difference,  because  this  may  be  obtained  directly  from  the 
American  Ephemeris,  and  is  at  the  same  time  even  more  accurate.  We  also  extend 
our  table  so  as  to  meet  the  case  where  one  altitude  is  taken  in  the  afternoon  and  the 
corresponding  equal  altitude  on  the  following  morning ;  in  which  case,  the  equation  is 
computed  for  apparent  midnight.* 


*  It  should  be  observed,  as  a  caution  to  navigators,  that  the  rule  for  computing  the  equation  for  mid- 
night is  sometimes  inaccurately,  or  incompletely,  stated  in  works  on  navigation  or  astronomy.  The 
rule  in  Lieut.  RAPER'S  Practice  of  Navigation  is  wholly  erroneous.  GALBRAITH'S  rule  (Mathematical 
and  Astronomical.  Tables)  is  incomplete,  in  not  noticing  the  case  where  the  elapsed  time  is  less  than  12^. 
His  rule  for  computing  the  equation  for  noon  is  similarly  defective,  in  not  noticing  the  case  where  the 
elapsed  time  is  greater  than  12h*.  In  Professor  INMAN'S  rule  there  is  a  slight  inaccuracy  introduced, 
by  taking  the  equation  of  time  for  mean,  instead  of  apparent  noon  or  midnight ;  and  in  all  the  books, 
9  65 


EQUAL    ALTITUDES. 

L    EQUAL  ALTITUDES   OF  THE   SUN,  MORNING  AND   EVENING. 

THE  OBSERVATION. 

On  shore,  at  a  place  whose  longitude  is  accurately  known,  and  whose  latitude  is 
approximately  known,  observe  with  an  artificial  horizon  the  same  altitude  both  in  the 
morning  and  in  the  afternoon,  as  near  the  prime  vertical  as  convenient  after  the  alti- 
tude is  more  than  10°,  noting  the  times  by  the  chronometer.  In  low  latitudes,  how- 
ever, the  method  of  equal  altitudes  will  often  give  very  accurate  results,  even  when 
the  observations  are  quite  near  to  the  meridian.  In  general,  a  sufficiently  accurate 
result  may  be  obtained  if  the  observations  are  taken  when  the  sun's  change  of  altitude 
is  not  less  than  10"  in  0"-.5,  or  when  the  change  in  the  double  altitude  taken  with  the 
artificial  horizon  is  not  less  than  20"  in  0s  .5. 

It  is  most  convenient,  as  well  as  conducive  to  accuracy,  to  take  the  observation  in 
the  following  manner.  In  the  morning,  bring  the  lower  limb  of  the  sun,  reflected 
from  the  sextant-mirrors,  and  the  upper  limb  of  that  reflected  from  the  mercury,  into 
approximate  contact ;  move  the  0  of  the  vernier  forward  (say  from  10'  to  20'),  and 
set  it  on  a  division  of  the  limb ;  the  images  will  be  overlapped  and  will  be  separating ; 
wait  for  the  instant  of  contact ;  note  it  by  chronometer,  and  immediately  set  the  ver- 
nier on  the  next  division  of  the  limb,  that  is,  10'  in  advance ;  note  the  instant  of  con- 
tact again,  and  proceed  in  the  same  manner  for  as  many  observations  as  are  thought 
necessary.  If  the  sun  rises  too  rapidly,  let  the  intervals  on  the  limb  be  20'.  Find 
(roughly)  the  time  when  the  sun  will  be  at  the  same  altitude  in  the  afternoon,  and 
just  before  that  time  set  the  vernier  on  the  last  altitude  noted  in  the  morning  (of 
course  using  the  same  sextant) ;  the  images  of  the  sun  will  be  separated,  but  will  be 
approaching ;  wait  for  the  instant  of  contact ;  note  it  by  chronometer ;  set  the  vernier 
back  to  the  next  division  of  the  limb  (10'  or  20',  as  the  case  may  be) ;  note  the  con- 
tact again,  and  so  proceed  till  all  the  A.  M.  altitudes  have  been  again  noted  as  P,  M. 
altitudes. 

THE  COMPUTATION. 

Take  the  mean  of  the  A.  M.  times  and  call  it  the  A.  M.  Chronometer  Time.  The 
mean  of  the  P.  M.  times  call  the  P.  M.  Chronometer  Time.  If,  instead  of  noting  the 
times  by  the  chronometer,  a  watch  is  used  (compared  with  the  chronometer  both 
before  and  after  each  observation),  it  will  generally  be  found  necessary  to  make  an 
allowance  for  its  gain  or  loss  on  the  chronometer,  so  as  to  obtain  the  exact  difference 
between  the  watch  and  chronometer  at  the  instant  of  observation.  This  difference 
being  applied  to  the  mean  of  the  watch  times,  we  have  the  mean  chronometer  time 
the  same  as  would  have  been  found  by  employing  the  chronometer  directly. 

the  methods  given  of  taking  out  the  sun's  change  of  declination,  whether  for  48ht  or  for  24h>,  are  not 
as  accurate  as  they  should  be. 

A  perfectly  accurate  rule,  with  a  special  table,  for  the  midnight  correction,  is  given  in  SCHUMACHER'S 
Hvlfstafeln  (Ed.  by  WAKNSTORFF).  It  requires,  however,  one  logarithm  more  than  our  method  in  the 
text,  and  is  otherwise  not  so  simple. 

66 


EQUAL    ALTITUDES. 

The  half  sum  of  the  A.  M.  and  P.  M.  Chronometer  Times  is  the  Middle  Chronom- 
eter Time,  their  difference  is  the  Elapsed  Time;  observing  that  when  the  A.  M. 
time  is  before  12h  by  chronometer,  while  the  P.  M.  time  is  after  12h-,  the  latter  must 
be  supposed  to  be  increased  by  12h  in  finding  this  half  sum  and  difference. 

Take  from  the  Nautical  Almanac  the  sun's  declination,  the  hourly  difference  of 
decimation,  and  the  equation  of  time,  reducing  each  to  the  instant  of  local  apparent 
noon  by  applying  the  changes  for  the  longitude. 

Mark  north  latitude  and  north  declination  •+- 
"     south  latitude  and  south  declination  — 
"     hourly  diff.  of  decl.  when  towards  north  •+• 
"     hourly  diff.  of  decl.  when  towards  south  — . 

Enter  Table  I.  with  the  elapsed  time,  and  take  out  log.  A  and  log.  B,  prefixing  to 
each  its  proper  sign  given  in  the  table  at  the  head  of  the  page. 

To  log.  A  add  the  log.  of  the  hourly  diff,  Table  II.,  and  the  log.  tangent  of  the 
latitude  (Bowditch,  Table  XXVII.).  Prefix  to  each  log.  the  sign  of  the  quantity  it 
represents  and  to  their  sum  the  sign  which  results  from  the  algebraic  combination  of 
the  three  signs.*  This  sum  is  the  log.  (Table  II.)  of  the  number  of  seconds  of  time 
in  the  first  part  of  equation  of  equal  attitudes,  to  be  marked  -f-  or  —  like  its  log. 

To  log.  B.  add  the  log.  of  the  hourly  diff.  and  the  log.  tangent  of  the  declination, 
marking  the  signs  as  before.  The  sum  is  the  log.  of  the  second  part  of  the  equation 
of  equal  altitudes,  to  be  marked  H-  or  —  like  its  log. 

Apply  the  two  parts  of  the  equation,  according  to  their  signs,  to  the  Middle  Chro- 
nometer Time  ;  the  result  is  the  Chronometer  Time  of  Apparent  Noon. 

To  this  apply  the  equation  of  time  (adding,  when  the  equation  of  time  is  additive 
to  mean  time,  otherwise  subtracting) ;  the  result  is  the  Chronometer  Time  of  Mean 
Noon,  which,  if  the  chronometer  is  regulated  to  local  time,  will  be  12h  Om>  0s-  when 
the  chronometer  is-  right ;  more  than  12ht  when  fast,  less  than  12h<  when  slow. 

If  the  chronometer  is  regulated  to  Greenwich  time,  apply  the  longitude  (in  time)  to 
the  chronometer  time  of  mean  noon  (subtracting  in  west,  adding  in  east) ;  the  result 
will  be  more  or  less  than  12h ,  according  as  the  chronometer  is  fast  or  slow. 

Repeat  this  process  on  a  subsequent  day.  The  difference  between  the  chronome- 
ter errors  on  the  two  days,  divided  by  the  number  of  days  in  the  interval,  is  the  daily 
rate  of  the  chronometer,  gaining  or  losing  according  as  the  chronometer  goes  too 
fast  or  too  slow. 

EXAMPLE  1. 

May  3d,  1856.  At  the  United  States  Naval  Academy,  Lat.  38°  59'  N.,  Long. 
5h*  5m-  55*  .1  W.,  suppose  the  following  observations  of  equal  altitudes  to  be  taken  with 
an  artificial  horizon.  Required  the  error  of  the  chronometer  on  Greenwich  time  at 
noon  of  that  day  ? 

*  The  algebraic  rule  being,  that,  when  there  is  an  odd  number  of  factors  with  the  sign  minus, 
the  result  must  have  the  sign  minus,  otherwise  the  sign  plus.  In  the  present  application  of  this 
rule,  when  there  is  either  one  or  three  of  the  logs,  marked  — ,  their  sum  must  be  marked  —  j 
otherwise  4-. 

67 


EQUAL    ALTITUDES. 


A.M. 

Comparisons. 

h.    m.      8. 

Ohronom.               12  52    0.0 
Watch                       7  45     8.0 
Diff.                          5    6  52.0 

A.  M.,  watch  gains  l':5  in  28m* 
Interval  to  obs.  17m%5 

28m.  .  17m.<5  __  p.  5  .  os%9 

P.  M.,  watch  gains  2'-.2  in  34m- 
Interval  to  obs.  21m< 
34m.  .  21m.  _  2s.>2  :  is.,4 

2  O  Art.  Hor. 

o     t 
65  50 
66     0 
66  10 

(Eq.  T.)  —  "'iS-ll                   0.258 
1.32                    5.1 

P.M. 

Comparisons, 
h.  m.     a. 
Chronom.                   8  37    0.0 
Watch                         3  30  31.3 

Diff.                              5     6  28.7 

Chronom.                  9  11    0.0 
Watch                         4    4  33.5 

Chronom.                 1  20    0.0 
Watch                       8  13     9.5 

Diff.                          5     6  50.5 

Diff.                            5     6  26.5 

Watch  A.  M. 
h.   m.    8. 
829. 
8     2  35.5 
8     3     0.5 

Watch  P.  M. 
h.  m.    s. 
3  52  10.7 
3  51  44.0 
3  51   18.5 

Mean                       8    2  35.0 
Comparison             5    6  51.1 

Mean                         3  51  44.4 
Comparison                5     6  27.3 
P.  M.  Chro.  Time     8  58  11.7 
A.  M.  Chro.  Time     1     9  26.1 
Elapsed  Time            7  48  45.6 

*•              V 

A.  M.  Chro.  Time   1     9  26.1 
P.  M.  Chro.  Time    8  58  11.7 

2)10    7  37.8 

Eq.  T.      —3  19.43                    1.32 

(D.)  +15  48  50.5      (H.  D.)  +43.82 
3  42.8                     —0.14 

Middle  Chro.  T.       53  48.9 
Equat.  of  Eq.  Alts.           —8.8 

Chro.  T.  App.  N.      5     3  40.1 
Equat.  of  Time          H-3  19.4 

Decrease  in   5.1  =  0.14 

log.  B.  Tab.  I.  +9.201! 
+1  6403 

D.     +155233.3       H.  D.  +43.68 
5.1 

Chro.  T.  Mean  N.     5    6  59.5 
Longitude                5    5  55.1  W. 

222.8 

log.A.Tab.I.  —9.4846 
H.D.+43".68  log.  Tab.  II.  +1.6403 
Lat.  +  38°  59'  log.  tan.         +9.9081 

Chro./orf                      1    4.4 

D.  +  15°  53'  log.  tan.  +9.4542 

1st  Pt.  Eq.  —  10*-.79     log.  —1.0330 

2d  Pt.  Eq.  +  lf  -.98  log.  +0.2956 

By  similar  observations  on  May  15th,  suppose  the  chronometer  is  found  to  be  fast 
I2§\5;  we  have 


May  3d,  fast 
May  15th,  fast 
Loses  in  12  days 
Daily  rate 


4.4 

12.5 
_51.9 
~4.33~  losing. 


H.  EQUAL  ALTITUDES  OF  THE  SUN,  EVENING  AND  MORNING. 

THE  OBSERVATION. 

Take  a  set  of  altitudes,  in  the  manner  already  explained,  in  the  afternoon  of  one 
day,  and  the  same  altitudes  in  reverse  order  on  the  morning  of  the  next,  noting  the 
times  by  the  chronometer,  or  by  a  watch  compared  with  it. 

THE  COMPUTATION. 

The  half  sum  of  the  P.  M.  and  A.  M.  Chronometer  Times  is  the  Middle  Chronom- 
eter Time ;  their  difference  is  the  Elapsed  Time ;  observing  that  when  the  P.  M. 
time  is  before  12h  by  chronometer,  while  the  A.  M.  time  is  after  12h ,  the  latter  must 
be  supposed  to  be  increased  by  12h-  in  finding  this  half  sum  and  this  difference. 

Take  from  the  Nautical  Almanac  the  sun's  declination,  the  hourly  difference  of 
declination,  and  the  equation  of  time,  reducing  them  each  to  the  instant  of  local  flp- 
parent  midnight. 

68 


EQUAL    ALTITUDES. 

Mark  the  sign  of  each  quantity  as  before,  and  compute  the  two  parts  of  the  equa- 
tion of  equal  altitudes  precisely  as  in  the  preceding  case,  observing  to  mark  the  signs 
of  log.  A  and  log.  B  as  given  in  the  table  for  midnight. 

Apply  the  two  parts  of  the  equation  to  the  middle  chronometer  time,  according  to 
their  signs ;  the  result  is  the  Chronometer  Time  of  Apparent  Midnight. 

To  this  apply  the  equation  of  time  (adding,  when  the  equation  of  time  is  additive 
to  mean  time,  otherwise  subtracting) ;  the  result  is  the  Chronometer  Time  of  Mean 
Midnight,  which,  if  the  chronometer  is  regulated  to  local  time,  will  be  12hi  Ora  O"1 
when  the  chronometer  is  right ;  more  than  12hl  when  fast ;  less  than  12h>  when  slow. 

If  the  chronometer  is  regulated  to  Greenwich  time,  apply  the  longitude,  in  time,  to 
the  chronometer  time  of  mean  midnight  (subtracting  in  west,  adding  in  east) ;  the  re- 
sult will  be  more  or  less  than  12h>  (or  24h-)  according  as  the  chronometer  is  fast  or  slow. 

A  repetition  of  this  process  at  a  subsequent  day  will  give  another  error,  whence  the 
rate  will  be  found  as  before.  Or  the  rate  may  be  found  by  comparing  the  results  of 
an  A.  M.  —  P.  M.,  and  a  P.  M.  —  A.  M.  observation,  remembering  that  the  interval 
elapsed  between  two  such  observations  is  equal  to  the  difference  between  the  two 
dates  plus  or  minus  half  a  day. 

EXAMPLE  2. 

May  3d,  1856,  Lat.  43°  21'  S.,  Long.  9h-  50m-  8s-  E.,  suppose  the  altitude  of  the 
sun  to  be  observed  in  the  afternoon  and  the  same  altitude  again  on  the  morning  of 
the  4th,  as  below.  Required  the  error  of  the  chronometer  on  Greenwich  time  at  mid- 
night of  the  3d  ? 

Chronom.,  P.  M.  2  ©Art.  Hor.  Chronom.,  A.  M. 

6h.  54m.  10s.3  ~38o  Q,  gh.  9m.  17s.5 

The  A.  M.  time  must  be  called  21h-  9m  178>.5.  The  Greenwich  time  of  midnight 
for  which  the  declination,  &c.  must  be  found,  is  May  3d-  2h-  9m>  52s-  (=  3d-  2h  .16.). 


P.  M.  Chro.  T.         6'  54  H).3 
A.M.  Chro.  T.       21     9  17.5 

m    s.                  s. 
(Eq.  T.)  —3  18.11             0.258 
0.56             2.15 

h.  m.    s. 
A.  M.  Chro.  T.  21     9  17.5 
P.  M.  Chro.  T.     6  54  10.3 

2)28     3  27.8 

Eq.T.      —3  18.67             0.56 

(D.)  +1°5  48  50.5     (H.  D.)  +43.82 
1  34.1                     —0.06 

Elapsed  T.          14  15     7.2 

h.            „ 
Decrease  in  24.0    =  0.66 
Decrease  in    2.16  =  0.06 

log.  B.  Tab.  I.  —9.1586 
.     .     .     .          .          +1  6411 

Middle  Chro.  T.     14    1  43.9 
Eq.  of  Eq.  Alts.              —22.3 

Chro.T.App.Midn.  14     1  21.6 
Eq.  of  Time                -1-3  18.7 

D.     +15  50  24.6       H.  D    +43.76 
2.15 

Chro.T.M'n  Midn.  14    440.3 
Longitude                 9  50    8.0  E. 
23  54^48.3 
24     0     0.0 

94.1 

log.A.Tab.L  +9.6958 
H.D.+43".76  log.  Tab.  II.  +1.6411 
Lat,  —43°  21'  log.  tan.        —9.9750 
1st  Pt.  Eq.  —  20*-.51      log.  —  T.s!l9 

Chronom.  slow               5  1  1  .7 

D  +  15°  50'  log.  tan.  +9.4527 
2dPt.Eq.  —  P-.79  log.  —  (X2524 

By  an  A.  M.  —  P.  M.  observation  on  May  20th,  suppose  this  chronometer  is  found 
to  be  slow  8m-  148>,6  ;  we  have 

d.    h.  m.  s. 

May    3  12    slow  5  11.7 

May  20    0    slow  8  14.6 

Loses  in  16d-.5  ~3~2.9~ 
Daily  rate 

69 


11.09  losing. 


EQUAL    ALTITUDES. 


III.     EQUAL  ALTITUDES   OF  A   FIXED   STAR. 

THE  OBSERVATION. 

In  selecting  stars  for  this  observation,  it  is  to  be  observed  that  the  nearer  the  zenitri 
the  star  passes,  the  less  may  the  elapsed  time  be ;  and  when  the  star  passes  exactly 
through  the  zenith,  the  two  altitudes  may  be  taken  within  a  few  minutes  of  each 
other.  But  with  the  ordinary  sextants,  altitudes  near  90°  cannot  be  taken  with  the 
artificial  horizon,  as  the  double  altitude  is  then  nearly  180°.  The  prismatic  sextants, 
or  still  better,  the  prismatic  circles  of  Pistor  and  Martin,  are  adapted  for  measuring 
angles  of  all  magnitudes  up  to  180°,  and  are  therefore  especially  suitable  for  this  ob- 
servation. 

Set  the  sextant  and  wait  for  the  coincidences  of  the  two  images  of  the  star,  as  in 
the  case  of  the  sun's  limb,  noting  the  times  by  chronometer  or  watch. 

THE  COMPUTATION. 

Take  the  mean  of  the  times  before  the  meridian  passage  as  the  A.  M.  Chronome- 
ter Time,  and  the  mean  of  those  after  the  meridian  passage  as  the  P.  M.  Chronome* 
ter  Time. 

The  mean  of  the  A.  M.  and  P.  M.  Chronometer  Times  is  the  Chronometer  Time  of 
Star's  Transit.  This  time,  if  the  chronometer  is  right,  will  agree  with  the  true  mean 
time  of  star's  transit,  which  is  to  be  computed  as  follows. 

To  the  right  ascension  of  the  star  apply  the  longitude  of  the  place  of  observation 
(adding  in  west,  subtracting  in  east) ;  the  result  is  the  Greenwich  Sidereal  Time  of 
Star's  Transit,  from  which  subtract  the  sidereal  time  at  the  preceding  mean  noon 
Greenwich  (Nautical  Almanac,  page  II.  of  the  month) ;  the  remainder  is  the  Sidereal 
Interval  since  mean  noon.  From  Table  IV.  with  the  argument  Sidereal  Interval,  take 
out  the  correction,  which  subtract  from  the  sidereal  interval ;  the  remainder  is  the 
Greenwich  Mean  Time  of  the  Star's  Transit.  The  chronometer  time  will  be  more 
or  less  than  this  according  as  the  chronometer  is  fast  or  slow. 

If  the  chronometer  is  regulated  to  local  time,  apply  the  longitude  to  the  Greenwich 
mean  time  of  star's  transit  (subtracting  in  west,  adding  in  east)  ;  the  result  is  the 
Local  Mean  Time  of  Star's  Transit,  and  the  chronometer  is  fast  or  slow  according  as 
it  shows  more  or  less  than  this  time. 

EXAMPLE  3. 

July  15th,  1856,  at  the  Cape  of  Good  Hope,  Lat.  33°  56' S.,  Long.  lh-  13re-  56s-  E., 
observed  equal  altitudes  of  Antares  as  follows :  — 


70 


EQUAL    ALTITUDES. 

Chronom.  A.  M.                    2  Alt.  Antares.  Chronom.  P.  M 

h-  m.     8.                                      o        i  h.  m.    s 

5  32  10.5                              125  30  9  34  20.3 

5  32  35.0                                      40  9  33  56.0 

5  32  59.3                                      50  9  33  32.0 

A.  M.  Chr..  T.  5  32  34.9      .  P.  M.  wnro.  T.    9  33  56.f 

P.  M.  Chro.  T.                  9  33  56.1  

2)15     6  31.0                              Antares  R.  A,  16  20  37.58 

Chro.  T.  #  Transit         7  33  15.5                             Longitude  1  13  56.00  E 

Gr.  T.  *  Transit             7  31  22.1                               Gr.  Sid.  T.  15     641.58 
Chro.  fast                              1  53.4                            July  15,  Gr.  Sid.  T.  Mean  Noon    7  34    5.25 

Sid.  Interval  7  32  36.33 

Correction,  Table  IV.  —1  14.15 

Gr.  M.  T.  >fc  Transit  7  31  22.18 


IV.    TO  CORRECT  FOR  SMALL  INEQUALITIES  IN  THE  ALTITUDES 

Although  the  sextant  readings  are  the  same  at  the  A.  M.  and  P.  M.  observations,  it 
may  happen  that  neither  the  true  nor  even  the  apparent  altitudes  are  the  same. 
1st.  Supposing  the  sextant  to  remain  unchanged,  the  atmospheric  refraction  may  be 
different  at  the  two  observations  in  consequence  of  changes  in  the  density  and  tem- 
perature of  the  air  as  shown  by  the  barometer  and  thermometer.  In  this  case,  the 
apparent  altitudes  are  equal,  but  the  true  altitudes  are  not  so.  2d.  The  sextant  may 
be  affected  by  changes  of  temperature,  particularly  in  day  observations  in  the  sun,  so 
as  to  make  the  sextant  readings  the  same  for  apparent  altitudes  slightly  different.  I 
do  not  think  these  changes  in  the  sextant  are  to  be  eliminated  by  determining  the 
index  error  at  each  observation,  as  has  been  supposed  by  some,  since  it  is  quite  pos- 
sible that  the  expansion  and  contraction  of  the  various  parts  might  leave  the  index 
correction  unchanged  while  it  affected  the  readings  of  the  altitudes,  or  the  reverse. 
The  only  course  appears  to  be  to  guard  the  instrument  as  much  as  possible  from 
changes  of  temperature,  exposing  it  to  the  sun's  rays  only  during  the  few  minutes  re- 
quired for  fiach  observation. 

But  the  correction  for  changes  of  refraction  may  be  satisfactorily  made  as  follows. 
Note  the  barometer  and  thermometer  both  A.  M.  and  P.  M. ;  take  out  the  correspond- 
ing refractions  for  each  observation  from  Tables  III.,  III.  A.,  and  III.  B.,  and  find  the 
difference  of  these  refractions.  Also  take  the  difference  between  any  two  sextant 
readings  and  the  difference  between  the  two  corresponding  chronometer  times.  Then 
the  correction  of  either  noon  or  midnight  will  be  found  by  the  following  proportion. 
The  difference  of  the  sextant  readings  is  to  the  difference  of  the  refractions  as  the  dif- 
ference of  the  chronometer  times  is  to  the  renuired  correction. 

Apply  this  correction  to  the  Chronometer  Time  of  Noon  or  Midnight  (obtained  by 
the  preceding  rules)  as  follows :  add  it  when  the  A.  M.  refraction  is  the  greater ; 
subtract  it  when  the  P.  M.  refraction  is  the  greater.  The  result  is  the  true  Chronom- 
eter Time  of  Noon  or  Midnight. 

EXAMPLE.  —  Suppose,  in  Example  1,  we  have  in  the  morning,  Barometer  30  inches, 
Thermometer  55° ;  in  the  afternoon,  Barometer  29.5  inches,  Thermometer  85°.  The 
apparent  altitude  of  sun's  lower  limb  33°  0' ;  the  apparent  altitude  of  sun's  centre 
33°  16'.  We  have 

71 


EQUAL    ALTITUDES. 

A  M.  P.  M. 

Mean  refraction  1  29  Mean  refraction  1  29 

Barom.  30  in.  0  Barorn.  29.5  in.  —1 

Therm.  55°  — 1  Therm.  85°  —6 

True  refraction  1  28  True  refraction  1  22 

Then  the  difference  of  the  sextant  readings  is  10'  (=600")  and  the  corresponding 
diff.  of  chronometer  times  is  about  26*  ;  whence 

II         U  8.        B 

600  :  6  =  26  :  0.26 

h.  m.  B. 

The  (approximate)  Chronometer  Time  of  Mean  Noon  was  found  to  be  5  6  59.5 
Correction  for  change  of  refraction  -1-0.3 

True  Chronometer  Time  of  Mean  Noon  5  6  59.8 

NOTE.  —  This  correction  may  be  found  by  the  following  rule,  which  we  should  have  to  resort  to 
when  but  one  altitude  was  taken  at  each  observation.  Add  together  the  log.  of  the  diff.  of  refractions 
(Tab.  II.),  log.  cosine  of  the  altitude,  log.  secant  of  the  latitude,  log.  secant  of  the  declination,  log. 
cosecant  of  half  elapsed  time  (or  if  the  elapsed  time  is  greater  than  12h>,  half  its  supplement  to  24h'), 
and  the  constant  log.  8.523 ;  the  sum  is  the  log.  (Table  II.)  of  the  required  correction.  Thus  in  the 
preceding  example  we  have 

Diff.refr.            6"  log.  0.778 

Alt.  0        33°  16'  log.  cos.  9.922 

Lat.             38°  59'  log.  sec.  0.109 

Dec.            15°  53'  log.  sec.  0.017 

El.  T.           7h-  49m>  log.  cosec.*  0.069 

const,  log.  8.523 

Correction    08\26  log.  9.418 

DEGREE  OF  DEPENDENCE. 

An  error  of  5'  in  the  latitude  would  not  affect  the  corresponding  part  of  the  equa- 
tion of  equal  altitudes  by  more  than  one  hundredth  of  its  amount  in  the  most  unfavor- 
able case,  and  in  general  would  have  no  sensible  effect.  It  is  one  of  the  advantages 
of  the  equal  altitude  method,  therefore,  that  it  does  not  require  an  accurate  knowledge 
of  the  latitude.  It  is  also  plain  that  errors  in  the  longitude  affecting  the  declination 
and  its  hourly  difference  produce  but  small  proportionate  effects  upon  the  computed 
equation.  The  absolute  error  of  the  chronometer  on  Greenwich  will  be  affected  by 
the  whole  error  in  the  longitude,  but  the  rate  will  still  be  correct.  Hence  we  con- 
clude that  by  this  method  the  chronometer  may  be  accurately  rated  at  a  place  whose 
latitude  and  longitude  are  both  imperfectly  known. 

The  chief  source  of  error  is  in  the  observation  itself.  The  most  practised  observers 
with  the  sextant  cannot  depend  on  the  noted  time  of  a  single  contact  within  0"  .5,  and 
hence  the  intervals  between  the  successive  chronometer  times  (which,  if  observations 
could  be  perfectly  taken  would  be  sensibly  equal)  may  differ  2" .  But  the  greatest 
probable  error  of  the  chronometer  time  of  sun's  or  star's  transit,  from  the  mean  of  six 
such  observations  on  each  side  of  the  meridian,  is  found  to  be  not  more  than  0s-. 2, 
provided  the  rate  of  the  chronometer  between  the  observations  is  uniform. 

Errors  resulting  from  changes  in  the  refraction  may  be  almost  wholly  removed  by 
computation  as  above. 

*  Enter  BOWDITCH'S  Table  XXVII.,  column  P.  M.,  with  the  whole  elapsed  time  and  take  out  the 
corresponding  cosecant. 

72 


EXPLANATION  OF  THE  TABLES 


TABLE  I.  —  Logarithms  of  A  and  B,  for  computing  the  Equation  of  Equal  Alfo 
tudes,  are  calculated  by  the  formulas 

A  -=          E  R  -        _E 

1800  sin  i  E  '  1800  tan  4  E  ' 

where  E  =  elapsed  time  in  minutes,  and  E  in  the  denominator  is  the  elapsed  time 
expressed  in  arc. 

If  we  put 

$  =  latitude  of  the  place  of  observation,  H-  north,  —  south, 
b  =  declination  of  the  sun,  H-  north,  —  south, 

A  =  hourly  change  of  declination,  •+-  north,  —  south, 

X  =  correction  to  reduce  the  middle  chronometer  time  to  chronometer  time  of  ap- 
parent noon,  algebraically  additive, 
tf  =  the  same  for  midnight, 

we  have 

X  =  —  A  A  tan  <f>  4-  B  A  tan  d 

XI  =       A  A  tan  <£  -+  B  A  tan  8. 

TABLE  II.  —  Logarithms  of  Numbers  to  four  decimal  places.  The  first  two  fig-* 
ures  of  the  number  are  found  in  the  left-hand  column,  the  third  at  the  top,  and  the 
corresponding  logarithm  opposite  and  under  these  respectively.  The  proportional 
part  for  the  fourth  figure  is  found  on  the  side  in  the  same  line  with  the  logarithm  taken 
out.  The  proper  characteristic  of  the  logarithm  is  to  be  supplied  by  the  usual  rule. 

TABLE  III.  —  Mean  Refraction,  reduced  from  BESSEL'S  Tables,  to  barometer  30 
inches,  and  thermometer  50°. 

TABLES  III.  A.  and  III.  B.  —  Corrections  of  the  Mean  Refraction  for  the  Height 
of  the  Barometer  and  Thermometer,  also  deduced  from  BESSEL'S  Tables.  These  are 
the  same  as  Tables  IV.  A.  and  IV.  B.,  given  in  the  Appendix  to  the  Nautical  Alma- 
nac for  1855,  where  they  are  used  for  finding  the  corrections  of  the  Mean  Reduced 
Refraction  for  Lunars.  It  is  for  the  purpose  of  having  the  same  table  for  correct- 
ing both  these  mean  refraction  tables,  that  the  argument  in  Tables  III.  A.  and  III.  B* 
is  the  mean  refraction  instead  of  the  apparent  altitude. 

TABLE  IV.  —  For  converting  Sidereal  into  Mean  Solar  Time.     This  table  gives 
the  correction  required  to  reduce  a  sidereal  interval  to  its  equivalent  solar  interval. 
10  73 


TABLE  I.    LOG.  A.  AND  LOG.  B. 

For  Computing  the  Equation  of  Equal  Altitudes. 


KSS3&TA+}     ARGUMENT  =  ELAPSED  TIME.       {  ^Lfg°h0t?B+' 

|l 

Qh. 

lh' 

r 

f- 

4h- 

£h. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

0 

9.4059 

9.4059 

9.4072 

9.4034 

9.4109 

9.3959 

9.4172 

9.3828 

9.4260 

9.3635 

9.4374 

9.3369 

i 

.4059 

.4059 

.4072 

.4034 

.4110 

.3957 

.4173 

.3825 

.4261 

.3631 

.4376 

.3364 

2 

.4059 

.4059 

.4073 

.4033 

.4111 

.3955 

.4174 

.3822 

.4263 

.3627 

.4378 

.3358 

3 

.4059 

.4059 

.4073 

.4032 

.4112 

.3953 

.4175 

.3820 

.4265 

.3624 

.4380 

.3353 

4 

.4059 

.4059 

.4074 

.4031 

.4113 

.3952 

.4177 

.3817 

.4266 

.3620 

.4383 

.3348 

5 

9.4059 

9.4059 

9.4074 

9.4030 

9.4113 

9.3950 

9.4178 

9.3814 

9.4268 

9.3616 

9.4385 

9.3S43 

6 

.4060 

.4059 

.4074 

.4029 

.4114 

.3948 

.4179 

.3811 

.4270 

.3612 

.4387 

.3337 

7 

.4060 

.4059 

.4075 

.4028 

.4115 

.3946 

.4181 

.3809 

.4272 

.3608 

.4389 

.3332 

8 

.4060 

.4059 

.4075 

.4027 

.4116 

.3944 

.4182 

.3806 

.4273 

.3604 

.4391 

.3327 

9 

.4060 

.4059 

.4076 

.4026 

.4117 

.3943 

.4183 

.3803 

.4275 

.3600 

.4393 

.3321 

10 

9.4060 

9.4059 

9.4076 

9.4025 

9.4118 

9.3941 

9.4184 

9.3800 

9.4277 

9.3596 

9.4396 

9.3316 

11 

.4060 

.4059 

.4077 

.4024 

.4119 

.3939 

.4186 

.3797 

.4279 

.3592 

.4398 

.3311 

12 

.4060 

.4058 

.4077 

.4023 

.4120 

.3937 

.4187 

.3794 

.4280 

.3588 

.4400 

.3305 

13 

.4060 

.4058 

.4078 

.4022 

.4121 

.3935 

.4188 

.3792 

.4282 

.3584 

.4402 

.3300 

14 

.4060 

.4058 

.4078 

.4021 

.4121 

.3933 

.4190 

.3789 

.4284 

.3580 

.4405 

.3294 

15 

9.4060 

9.4058 

9.4079 

9.4020 

9.4122 

9.3931 

9.4191 

9.3786 

9.4286 

9.3576 

9.4407 

9.3289 

16 

.4060 

.4058 

.4079 

.4019 

.4123 

.3929 

.4193 

.3783 

.4288 

.3572 

.4409 

.3283 

17 

.4060 

.4057 

.4080 

.4018 

.4124 

.3927 

.4194 

.3780 

.4289 

.3568 

.4411 

.3278 

18 

.4061 

.4057 

.4080 

.4017 

.4125 

.3925 

.4195 

.3777 

.4291 

.3564 

.4414 

.3272 

19 

.4061 

.4057 

.4081 

.4016 

.4126 

.3923 

.4197 

.3774 

.4293 

.3559 

.4416 

.3266 

20 

9.4061 

9.4057 

9.4081 

9.4015 

9.4127 

9.3921 

9.4198 

9.3771 

9.4295 

9.3555 

9.4418 

9.3261 

21 

.4061 

.4056 

.4082 

.4014 

.4128 

.3919 

.4199 

.3768 

.4297 

.3551 

.4420 

.3255 

22 

.4061 

.4056 

.4083 

.4013 

.4129 

.3917 

.4201 

.3765 

.4299 

.3547 

.4423 

.3249 

23 

.4061 

.4056 

.4083 

.4012 

.4130 

.3915 

.4202 

.3762 

.4300 

.3542 

.4425 

.3244 

24 

.4061 

.4055 

.4084 

.4010 

.4131 

.3913 

.4204 

.3759 

.4302 

.3538 

.4427 

.3238 

25 

9.4062 

9.4055 

9.4084 

9.4009 

9.4132 

9.3911 

9.4205 

9.3756 

9.4304 

9.3534 

9.4430 

9.3232 

26 

.4062 

.4055 

.4085 

.4008 

.4133 

.3909 

.4207 

.3752 

.4306 

.3530 

.4432 

.3226 

27 

.4062 

.4054 

.4086 

.4007 

.4134 

.3907 

.4208 

.3749 

.4308 

.3525 

.4434 

.3220 

28 

.4062 

.4054 

.4086 

.4006 

.4135 

.3905 

.4209 

.3746 

.4310 

.3521 

.4437 

.3214 

29 

.4062 

.4054 

.4087 

.4004 

.4136 

.3903 

.4211 

.3743 

.4312 

.3516 

.4439 

.3208 

30 

9.4062 

9.4053 

9.4087 

9.4003 

9.4137 

9.3900 

9.4212 

9.3740 

9.4314 

9.3512 

9.4441 

9.3203 

31 

.4063 

.4053 

.4088 

.4002 

.4138 

.3898 

.4214 

.3737 

.4315 

.3508 

.4444 

.3197 

32 

.4063 

.4052 

.4089 

.4001 

.4139 

.3896 

.4215 

.3733 

.4317 

.3503 

.4446 

.3191 

33 

.4063 

.4052 

.4089 

.3999 

.4140 

.3894 

.4217 

.3730 

.4319 

.3499 

.4448 

.3185 

34 

.4063 

.4051 

.4090 

.3998 

.4141 

.3892 

.4218 

.3727 

.4321 

.3494 

.4451 

.3178 

35 

9.4064 

9.4051 

9.4091 

9.3997 

9.4142 

9.3889 

9.4220 

9.3723 

9.4323 

9.3490 

9.4453 

9.3172 

36 

.4064 

.4050 

.4091 

.3995 

.4144 

.3887 

.4221 

.3720 

.4325 

.3485 

.4456 

.3166 

37 

.4064 

.4050 

.4092 

.3994 

.4145 

.3885 

.4223 

.3717 

.4327 

.3480 

.4458 

.3160 

38 

.4064 

.4049 

.4093 

.3993 

.4146 

.3882 

.4224 

.3713 

.4329 

.3476 

.4460 

.3154 

39 

.4065 

.4049 

.4093 

.3991 

.4147 

.3880 

.4226 

.3710 

.4331 

.3471 

.4463 

.3148 

40 

9.4065 

9.4048 

9.4094 

9.3990 

9.4148 

9.3878 

9.4227 

9.3707 

9.4333 

9.3467 

9.4465 

9.3142 

41 

.4065 

.4048 

.4095 

.3988 

.4149 

.3875 

.4229 

.3703 

.4335 

.3462 

.4468 

.3135 

42 

.4065 

.4047 

.4095 

.3987 

.4150 

.3873 

.4231 

.3700 

.4337 

.3457 

.4470 

.3129 

43 

.4066 

.4047 

.4096 

.3985 

.4151 

.3871 

.4232 

.3696 

.4339 

.3453 

.4473 

.3123 

44 

.4066 

.4046 

.4097 

.3984 

.4152 

.3868 

.4234 

.3693 

.4341 

.3448 

.4475 

.3116 

45 

9.4066 

9.4045 

9.4097 

9.3982 

9.4154 

9.3866 

9.4235 

9.3690 

9.4343 

9.3443 

9.4477 

9.3110 

46 

.4067 

.4045 

.4098 

.3981 

.4155 

.3863 

.4237 

.3686 

.4345 

.3438 

.4480 

.3103 

47 

.4067 

.4044 

.4099 

3979 

.4156 

.3861 

.4238 

.3683 

.4347 

.3433 

.4482 

.3097 

48 

.4067 

.4043 

.4100 

.3978 

.4157 

.3859 

.4240 

.3679 

.4349 

.3429 

.4485 

.3091 

49 

.4068 

.4043 

.4100 

.3976 

.4158 

.3856 

.4242 

.3675 

.4351 

.3424 

.4487 

.3084 

50 

9.4068 

9.4042 

9.4101 

9.3975 

9.4159 

9.3854 

9.4243 

9.3672 

9.4353 

9.3419 

9.4490 

9.3078 

51 

.4068 

.4041 

.4102 

.3973 

.4161 

.3851 

.4245 

.3668 

.4355 

.3414 

.4492 

.3071 

52 

.4069 

.4041 

.4103 

.3972 

.4162 

.3849 

.4246 

.3665 

.4357 

.3409 

.4494 

.3064 

53 

.4069 

.4040 

.4103 

.3970 

.4163 

.3846 

.4248 

.3661 

.4359 

.3404 

.4497 

.3058 

54 

.4069 

.4039 

.4104 

.3969 

.4164)  .3843 

.4250 

.3657 

.4361 

.3399 

.4500 

.3051 

55 

9.4070 

9.4038 

9.4105 

9.3967 

9.4165 

9.3841 

9.4251 

9.3654 

9.4363 

9.3394 

9.4503 

9.3044 

56 

.4070 

.4038 

.4106 

.3965 

.4167 

.3838 

.425a 

.3650 

.4366 

.3389 

.4505   .3038 

57 

.4071 

.4037 

.4107 

.3964 

.4168 

.3836 

.4255 

.3646 

.4368 

.3384 

.4508 

.3031 

58 

.4071 

.4036 

.4107 

.3962 

.4169'  .3833 

.4256 

.3643 

.4370 

.3379 

.4510 

.3024 

59 

.4071 

.4035 

.4108 

.3960 

.4170   .3830 

.4258 

.3639 

.4372 

.3374 

.4513 

.3017 

60 

9.4072 

94034 

9.4109 

9.3959 

9.4172  9.3828 

9.4260  9.3635 

9.4374 

9.3369 

9.4515:  9.3010 

74 


TABLE  I.    LOG.  A.  AND  LOG.  B 

For  Computing  the  Equation  of  Equal  Altitudes. 


forSgtt7A+}     ARGUMENT  =  ELAPSED  TIME.       {Miam*£?B+ 

1  s 

o<  a 

Be 

gh. 

7b. 

8h- 

gh. 

iob- 

Uh. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.B. 

m 
0 

9.4515 

9.3010 

9.4685 

9.2530 

9.4884 

9.1874 

9.5115 

9.0943 

95379 

8.9509 

9.5680 

8.6837 

1 

.4518 

.3003 

.4688 

.2520 

.4888 

.1861 

.5119 

.0925 

.5384 

.9478 

.5685 

.6770 

2 

.4521 

.2996 

.4691 

.2511 

.4892 

.1848 

.5123 

.0906 

.5389 

.9447 

.5691 

.6701 

3 

.4523 

.2989 

.4694 

.2502 

.4895 

.1835 

.5127 

.0887 

.5393 

.9416 

.5696 

.6632 

4 

.4526 

.2982 

.4697 

.2492 

.4899 

.1822 

.5132 

.0867 

.5398 

.9384 

.5701 

.6560 

5 

9.4528 

9.2975 

9.4701 

9.2483 

9.4902 

9.1809 

9.5136 

9.0848 

9.5403 

8.9352 

9.5707 

8.6488 

6 

.4531 

.2968 

.4704 

.2473 

.4906 

.1796 

.5140 

.0828 

.5408 

.9320 

.5712 

.6414 

7 

.4534 

.2961 

.4707 

.2463 

.4910 

.1782 

.5144 

.0809 

.5412 

.9287 

.5718 

.6339 

8 

.4536 

.2954 

.4710 

.2454 

.4913 

.1769 

.5148 

.0789 

.5417 

.9254 

.5723 

.6262 

9 

.4539 

.2947 

.4713 

.2444 

.4917 

.1756 

.5153 

.0769 

.5422 

.9221 

.5728 

.6183 

10 

9.4542 

9.2940 

9.4716 

9.2434 

9.4921 

9.1742 

9.5157 

9.0749 

9.5427 

8.9187 

9.5734 

8.6103 

11 

.4544 

.2932 

.4719 

.2425 

.4924 

.1728 

.5161 

.0729 

.5432 

.9153 

.5739 

.6021 

12 

.4547 

.2925 

.4723 

.2415 

.4928 

.1715 

.5165 

.0708 

.5436 

.9118 

,5745 

.5937 

13 

.4550 

.2918 

.4726 

.2405 

.4932 

.1701 

.5169 

.0688 

.5441 

.9083 

.5750 

.5852 

14 

.4552 

.2911 

.4729 

.2395 

.4935 

.1687 

.5174 

.0667 

.5446 

.9048 

.5756 

.5764 

15 

9.4555 

9.2903 

9.4732 

9.2385 

9.4939 

9.1673 

9.5178 

9.0646 

9.5451 

8.9013 

9.5761 

8.5674 

16 

.4558 

.2896 

.4735 

.2375 

.4943 

.1659 

.5182 

.0625 

.5456 

.8977 

.5767 

.5583 

17 

.4561 

.2888 

.4738 

.2365 

.4946 

.1645 

.5186 

.0604 

.5461 

.8940 

.5772 

.5488 

18 

.4563 

.2881 

.4742 

.2355 

.4950 

.1630 

.5191 

.0583 

.5466 

.8903 

.5778 

.5392 

19 

.4566 

.2873 

.4745 

.2344 

.4954 

.1616 

.5195 

.0561 

.5470 

.8866 

.5783 

.5293 

20 

9.4569 

9.2866 

9.4748 

9.2334 

9.4958 

9.1602 

9.5199 

9.0540 

9.5475 

8.8829 

9.5789 

8.5192 

21 

.4572 

.2858 

.4751 

.2324 

.4961 

.1587 

.5204 

.0518 

.5480 

.8791 

.5794 

.5088 

22 

.4574 

.2850 

.4755 

.2313 

.4965 

.1573 

.5208 

.0496 

.5485 

.8752 

.5800 

.4981 

23 

.4577 

.2843 

.4758 

.2303 

.4969 

.1558 

.5212 

.0474 

.5490 

.8713 

.5806 

.4871 

24 

.4580 

.2835 

.4761 

.2292 

.4973 

.1543 

.5217 

.0452 

.5495 

.8674 

.5811 

.4758 

25 

9.4583 

9,2827 

9.4764 

9.2282 

9.4977 

9.1528 

9.5221 

9.0429 

9.5500 

8.8634 

9.5817 

8.4641 

26 

.4585 

.2819 

.4768 

.2271 

.4980 

.1513 

.5225 

.0406 

.5505 

.8594 

.5822 

.4521 

27 

.4588 

.2812 

.4771 

.2261 

.4984 

.1498 

.5230 

.0383 

.5510 

.8553 

.5828 

.4397 

28 

.4591 

.2804 

.4774 

.2250 

.4988 

.1483 

.5234 

.0360 

.5515 

.8512 

.5834 

.4270 

29 

.4594 

.2796 

.4778 

.2239 

.4992 

.1468 

.5238 

.0337 

.5520 

.8470 

.5839 

.4138 

30 

9.4597 

9.2788 

9.4781 

9.2228 

9.4996 

9.1453 

9.5243 

9.0314 

9.5525 

8.8427 

9.5845 

8.4001 

31 

.4600 

.2780 

.4784 

.2217 

.5000 

.1437 

.5217 

.0290 

.5530 

.8384 

.5851 

.3860 

32 

.4602 

.2772 

.4788 

.2206 

.5003 

.1422 

.5252 

.0266 

.5535 

.8341 

.5856 

.3713 

33 

.4605 

.2764 

.4791 

.2195 

.5007 

.1406 

.5256 

.0242 

.5540 

.8297 

.5862 

.3561 

34 

.4608 

.2756 

.4794 

.2184 

.5011 

.1390 

.5261 

.0218 

.5545 

.8253 

.5868 

.3403 

35 

9.4611 

9.2747 

9.4798 

9.2173 

9.5015 

9.1375 

9.5265 

9.0194 

9.5550 

8.8208 

9.5874 

8.3239 

36 

.4614 

.2739 

.4801 

.2162 

.5019 

.1359 

.5269 

.0169 

.5555 

.8162 

.5879 

.3067 

37 

.4617 

.2731 

.4804 

.2151 

.5023 

.1343 

.5274 

.0144 

.5560 

.8115 

.5885 

.2888 

38 

.4620 

.2723 

.4808 

.2140 

.5027 

.1327 

.5278 

.0119 

.5565 

.8068 

.5891 

.2701 

39 

.4622 

.2714 

.481  1 

.2128 

.5031 

.1310 

.5283 

.0094 

.5570 

.8020 

.5897 

.2505 

40 

9.4625 

9.2706 

9.4815 

9.2117 

9.5035 

9.1294 

9.5287 

9.0069 

9.5576 

8.7972 

9.5902 

8.2299 

41 

.4628 

.2698 

.4818 

.2105 

.5038 

.1278 

.5292 

.0043 

.5581 

.7923 

.5908 

.2082 

42 

.4631 

.2689 

.4821 

.2094 

.5042 

.1261 

.5296 

.0017 

.5586 

.7873 

.5914 

.1853 

43 

.4634 

.2681 

.4825 

.2082 

.5046 

.1244 

.5301 

8.9991 

.5591 

.7823 

.5920 

.1611 

44 

.4637 

.2672 

.4828 

.2070 

.5050 

.1228 

.5305 

.9965 

.5596 

.7772 

.5926 

.1354 

45 

9.4640 

9.2664 

9.4832 

9.2059 

9.5054 

9.1211 

9.5310 

8.9938 

9.5601 

8.7720 

9.5931 

8.1080 

46 

.4643 

.2655 

.4835 

.2047 

.5058 

.1194 

.53  !  5 

.9911 

.5606 

.7668 

.5937 

.0786 

47 

.4646 

.2646 

.4839 

.2035 

.5062 

.1177 

.5319 

.9884 

.5612 

.7614 

.5943 

.0470 

48 

.4649 

.2638 

.4842 

.2023 

.5066 

.1159 

.5324 

.9857 

.5617 

.7560 

.5949 

.0128 

49 

.4652 

.2629 

.4846 

.2011 

.5070 

.1142 

.5328 

.9830 

.5622 

.7505 

.5955 

7.9756 

50 

9.4655 

9.2620 

9.4849 

9.1999 

9.5074 

9.1125 

9.5333  89802 

9.5827 

8.7449 

9.5961 

7.9348 

51 

.4658 

.2611 

.4853 

.1987 

.5078 

.1107 

.5337   .9774 

.5632 

.7392 

.5967 

.8897 

52 

.4661 

.2602 

.4856 

.1974 

.5082 

.1089 

.53421  .9745 

.5638 

.7335 

.5973 

.8391 

53 

.4664 

.2593 

.4860 

.1962 

.50861  .1072 

.5347   .9717 

.5643 

.7276 

.5979 

.7817 

54 

.4667 

.2584 

.4863 

.1950 

.5091  j  .1054 

.5351   .9688 

.5648 

.7217 

.5985 

.7154 

55 

9.4670 

9.2575 

9.4867 

9.1937 

9.5095  9.1036 

9.5356  8.9659 

9.5654 

8.7156 

9.5991 

7.6368 

56 

.4673 

.2566 

.4870 

.1925 

.5099 

.1017 

.5361)  .9630 

.5659 

.7094 

.5997 

.5405 

57 

.4676 

.2557 

.4874 

.1912 

.5103 

.0999 

.5365   .9600 

.5664 

.7032 

.6003 

.4162 

58 

.4679 

.2548 

.4877 

.1900 

.5107   .0981 

.5370   .9570 

.5669 

.6968 

.6009 

.2407 

59 

.4682 

.2539 

.4881)  .1887 

.51111  .0962 

.5375  -  .9540 

.5675 

.6903 

.6015 

6.9591 

60 

9.4685 

92530 

9.4884  9.1874 

9.5115  9.0943 

9.5379  8.9509 

9.5680 

8.6837 

9.6021 

/n/. 

75 


TABLE  1.    LOG.  A.  Ai\D  LOG. 

For  Computing  the  Equation  of  Equal  Altitudes. 


foJ  SSSgttTA+  }     ARGUMENT  =  ELAPSED  TIME.       {  Sdnfg^B- 

Elapsed 

Time. 

If" 

If" 

14h- 

ish- 

16"' 

1?h. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.B 

1 
Log.  A. 

Log.B. 

Log.  A. 

Log.B. 

Log.  A. 

Log.B. 

Log.  A. 

Log.  B. 

m 
0 

9.6021 

Inf. 

9.6406 

8.7563 

9.6841 

9.0971 

9.7333 

9.3162 

9.7895 

9.4884 

9.8539 

9.6383 

1 

.6027 

6.9603 

.6412 

.7641 

,6848 

.1014 

.7342 

.3194 

.7905 

.4911 

.8550 

.6407 

2 

.6033 

7.2431 

.6419 

.7718 

.6856 

.1057 

.7351 

.3225 

.7915 

.4937 

.8562 

.6431 

3 

.6039 

.4198 

.6426 

.7794 

.6864 

.1099 

.7360 

.3256 

.7925 

.4963 

.8573 

.6455 

4 

.6045 

.5453 

.6433 

.7868 

.6872 

.1141 

.7369 

.3287 

.7935 

.4990 

.8585 

.6478 

5 

9.6051 

7.6428 

9.6440 

8.7942 

9.6879 

9.1183 

9.7378 

9.3319 

9.7945 

9.5016 

9.8597 

9.6502 

6 

.6057 

.7226 

.6447 

.8015 

.6887 

.1224 

.7386 

.3350 

.7955 

.5042 

.8608 

.6526 

7 

.6063 

.7902 

.6454 

.8087 

.6895 

.1265 

.7395 

.3380 

.7965 

.5068 

.8620 

.6550 

8 

.6069 

.8488 

.6461 

.8158 

.6903 

.1306 

.7404 

.3411 

.7975 

.5094 

.8632 

.6573 

9 

.6075 

.9005 

.6467 

.8227 

.6911 

.1347 

.7413 

.3442 

.7986 

.5120 

.8644 

.6597 

10 

9.6082 

7.9469 

9.6474 

8.8296 

9.6919 

9.1387 

9.7422 

Q.3472 

9.7996 

9.5146 

9.8655 

9.6621 

11 

.6088 

.9889 

.6481 

.8364 

.6926 

.1428 

.7431 

.3503 

.8006 

.5171 

.8667 

.6644 

12 

.6094 

8.0273 

.6488 

.8432 

.6934 

.1468 

.7440 

.3533 

.8016 

.5197 

.8679 

.6668 

13 

.6100 

.0627 

.6495 

.8498 

.6942 

.1507 

.7449 

.3563 

.8027 

.5223 

.8691 

.6691 

14 

.6106 

.0955 

.6502 

.8564 

.6950 

.1547 

.7458 

.3593 

.8037 

.5248 

.8703 

.6715 

15 

9.6112 

8.1260 

9.6509 

8.8628 

9.6958 

9.1586 

9.7467 

9.3623 

9.8047 

9.5274 

9.8715 

9.6738 

16 

.6119 

.1547 

.6516 

.8692 

.6966 

.1625 

.7476 

.3653 

.8058 

.5300 

.8727 

.6762 

17 

.6125 

.1816 

.6523 

.8756 

.6974 

.1664 

.7485 

.3683 

.8068 

.5325 

.8739 

.6785 

18 

.6131 

.2071 

.6530 

.8818 

.6982 

.1703 

.7494 

.3713 

.8078 

.5351 

.8751 

.6809 

19 

.6137 

.2312 

.6538 

.8880 

.6990 

.1741 

.7503 

.3742 

.8089 

.5376 

.8763 

.6832 

20 

9.6144 

8.2541 

9.6545 

8.8941 

9.6998 

9.1779 

9.7512 

9.3772 

9.8099 

9.5401 

9.8775 

9.6856 

21 

.6150 

.2759 

.6552 

.9002 

.7006 

.1817 

.7522 

.3801 

.8110 

.5427 

.8787 

.6879 

22 

.6156 

.2967 

.6559 

.9062 

.7014 

.1855 

.7531 

.3831 

.8120 

.5452 

.8799 

.6903 

23 

.6163 

.3166 

.6566 

.9121 

.7022 

.1893 

.7540 

.3860 

.8131 

.5477 

.8812 

.6926 

24 

.6169 

.3357 

.6573 

.9180 

.7030 

.1930 

.7549 

.3889 

.8141 

.5502 

.8824 

.6949 

25 

9.6175 

8.3540 

9.6580 

8.9238 

9.7038 

9.1967 

9.7558 

9.3918 

9.8152 

9.5528 

9.8836 

9.6973 

26 

.6182 

.3717 

.6588 

.9295 

.7047 

.2004 

.7568 

.3947 

.8162 

.5553 

.8848 

.6996 

27 

.6188 

.3887 

.6595 

.9352 

.7055 

.2041 

.7577 

.3976 

.8173 

.5578 

.8861 

.7019 

28 

.6194 

.4051 

.6602 

.9408 

.7063 

.2078 

.7586 

.4005 

.8184 

.5603 

.8873 

.7043 

29 

.6201 

.4210 

.6609 

.9464 

.7071 

.2114 

.7595 

.4033 

.8194 

.5628 

.8885 

.7066 

30 

9.6207 

8.4363 

9.6616 

8.9519 

9.7079 

9.2150 

9.7605 

9.4062 

9.8205 

9.5653 

9.8898 

9.7089 

31 

.6214 

.4512 

.6624 

.9573 

.7088 

.2186 

.7614 

.4090 

.8216 

.5677 

.8910 

.7112 

32 

.6220 

.4657 

.6631 

.9627 

.7096 

.2222 

.7624 

.4119 

.8227 

.5702 

.8923 

.7136 

33 

.6226 

.4796 

.6638 

.9681 

.7104 

.2258 

.7633 

.4147 

.8237 

.5727 

.8935 

.7159 

34 

.6233 

.4932 

.6645 

.9734 

.7112 

.2293 

.7642 

.4175 

.8248 

.5752 

.8948 

.7182 

35 

9.6239 

8.5064 

9.6653 

8.9787 

9.7121 

9.2329 

9.7652 

9.4204 

9.8259 

9.5777 

9.8961 

9.7205 

36 

.6246 

.5192 

.6660 

.9839 

.7129 

.2364 

.7661 

.4232 

.8270 

.5801 

.8973 

.7228 

37 

.9252 

.5318 

.6667 

.9891 

.7137 

.2399 

.7671 

.4260 

.8281 

.5826 

.8986 

.7251 

38 

.6259 

.5440 

.6675 

.9942 

.7146 

.2434 

.7680 

.4288 

.8292 

.5850 

.8999 

.7275 

39 

.6265 

.5559 

.6682 

.9993 

.7154 

.2468 

.7690 

.4316 

.8303 

.5875 

.9011 

.7298 

40 

9.6272 

8.5675 

9.6690 

9.0043 

9.7162 

9.2503 

9.7699 

9.4343 

9.8314 

9.5900 

9.9024 

9.7321 

41 

.6279 

.5788 

.6697 

.0093 

.7171 

.2537 

.7709 

.4371 

.8325 

.5924 

.9037 

.7344 

42 

.6285 

.5899 

.6704 

.0142 

.7179 

.2571 

.7718 

.4399 

.8336 

.5948 

.9050 

.7367 

43 

.6292 

.6008 

.6712 

.0191 

.7187 

.2605 

.7728 

.4426 

.8347 

.5973 

.9063 

.7390 

44 

.6298 

.6114 

.6719 

.0240 

.7196 

.2639 

.7738 

.4454 

.8358 

.5997 

.9075 

.7413 

45 

9.6305 

8.6218 

9.6727 

9.0288 

9.7204 

9.2673 

9.7747 

9.4481 

9.8369 

9.6022 

9.9088 

9.7436 

46 

.6311 

.6320 

.6734 

.0336 

.7213 

.2706 

.7757 

.4509 

.8380 

.6046 

.9101 

.7459 

47 

.6318 

.6419 

.6742 

.0384 

.7221 

.2740 

.7767 

.4536 

.8391 

.6070 

.9114 

.7482 

48 

.6325 

.6517 

.6749 

.0431 

.7230 

.2773 

.7776 

.4563 

.8402 

.6094 

.9127 

.7505 

49 

.6331 

.6613 

.6757 

.0478 

.7238 

.2806 

.7786 

.4590 

.8414 

.6119 

.9140 

.7529 

50 

9.6338 

8.6707 

9.6764 

9.0524 

9.7247 

9.2839 

9.7796 

9.4617 

9.8425 

9.6143 

9.9154 

9.7552 

51 

.6345 

.6799 

.6772 

.0570 

.7256 

.2872 

.7806 

.4644 

.8436 

.6167 

.9167 

.7575 

52 

.6351 

.6890 

.6779 

.0616 

.7264 

.2905 

.7815 

.4671 

.8447 

.6191 

.9180 

.7598 

53 

.6358 

.6979 

.6787 

.0662 

.7273 

.2937 

.7825 

.4698 

.8459 

.6215 

.9193 

.7621 

54 

.6365 

.7067 

.6795 

.0707 

.7281   .2970 

.7835 

.4725 

.8470 

.6239 

.9206 

.7644 

55 

9.6372 

8.7153 

9.6802 

9.0752 

9.7290  9.3002 

9.7845 

9.4752 

9.8481 

9.6263 

9.9220 

9.7667 

56 

.6378 

.7237 

.6810 

.0796 

.7299 

.3034 

.7855 

.4778 

.8493 

.6287 

.9233 

.7690 

57 

.6385 

.7321 

.6818 

.0840 

.7307 

.3066 

.7865 

.4805 

.8504 

.6311 

.9246 

.7713 

58 

.6392 

.7402 

.6825 

.0884 

.7316 

.3098 

.7875 

.4831 

.8516 

.6335 

.9260   .7736 

59 

.6399 

.7483 

.6833 

.0928 

.7324 

.3130 

.7885 

.4858 

.8527 

.6359 

.9273  .7759 

60 

9.6406 

8  7563 

9.68411  9.0971 

9.7333  9.3162 

9.7895  9.4884 

9.8539 

9.6383 

9.9287  9.7782 

76 


TABLE  I.    LOG.  A.  AND  LOG.  B, 

For  Computing  the  Equation  of  Equal  Altitudes. 


lorSSgttTA-f}     ARGUMENT  =  ELAPSED  TIME.       {SSSi5?B- 

!| 

SH 

18h- 

19h- 

20h- 

21h- 

2f- 

ir 

Log.  A. 

Log.  B. 

Log.  A. 

Log.B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

Log.  A. 

Log.  B. 

m 
0 

9.9287 

9.7782 

0.0172 

9.9167 

0.1249 

0.0625 

0.2623 

0.2279 

04523 

0.4372 

0.7689 

0.7652 

1 

.9300 

.7804 

.0188 

.9190 

.1269 

.0650 

.2649 

.2309 

.4562 

.4414 

.7765 

.7729 

2 

.9314 

.7827 

.0204 

.9213 

.1290 

.0676 

.2676 

.2339 

.4601 

.4455 

.7842 

.7807 

3 

.9327 

.7850 

.0221 

.9237 

.1310 

.0701 

.2702 

.2370 

.4640 

.4497 

.7920 

.7886 

4 

.9341 

.7873 

.0237 

.9260 

.1330 

.0727 

.2729 

.2401 

.4680 

.4540 

.8000 

.7967 

5 

9.9355 

9.7896 

0.0253 

9.9284 

0.1351 

0.0753 

0.2756 

0.2431 

0.4720 

0.4582 

0.8081 

0.8049 

6 

.9368 

.7919 

.0270 

.9307 

.1371 

.0779 

.2783 

.2462 

.4761 

.4625 

.8163 

.8133 

7 

.9382 

.7942 

.0286 

.9331 

.1392 

.0805 

.2810 

.2493 

.4801 

.4668 

.8247 

.8218 

8 

.9396 

.7965 

.0303 

.9355 

.1412 

.0830 

.2838 

.2524 

.4842 

.4711 

.8333 

.8305 

9 

.9410 

.7988 

.0319 

.9378 

.1433 

.0856 

.2865 

.2556 

.4884 

.4755 

.8420 

.8393 

10 

9.9424 

9.8011 

0.0336 

9-9402 

0.1454 

0.0882 

0.2893 

0.2587 

0.4926 

0.4799 

0.8508 

0.8483 

11 

.9437 

.8034 

.0353 

.9426 

.1475 

.0909 

.2921 

.2619 

.4968 

.4844 

.8599 

.8574 

12 

.9451 

.8057 

.0370 

.9449 

.1496 

.0935 

.2949 

.2650 

.5010 

.4889 

.8691 

.8667 

13 

.9465 

.8080 

.0386 

.9473 

.1517 

.0961 

.2977 

.2682 

.5053 

.4934 

.8786 

.8763 

14 

.9479 

.8103 

.0403 

.9497 

.1538 

.0987 

.3005 

.2714 

.5097 

.4980 

.8882 

.8860 

15 

9.9493 

9.8126 

0.0420 

9.9520 

0.1559 

0.1013 

0.3034 

0.2746 

0.5140 

0.5026 

0.8980 

0.8959 

16 

.9508 

.8149 

.0437 

.9544 

.1581 

.1040 

.3063 

.2778 

.5184 

.5072 

.9080 

.9060 

17 

.9522 

.8172 

.0454 

.9568 

.1602 

.1066 

.3091 

.2811 

.5229 

.5118 

.9183 

.9164 

18 

.9536 

.8195 

.0472 

.9592 

.1623 

.1093 

.3120 

.2843 

.5274 

.5165 

.9288 

.9270 

19 

.9550 

.8218 

.0489 

.9616 

.1645 

.1119 

.3150 

.2876 

.5319 

.5213 

.9396 

.9378 

20 

9.9564 

9.8241 

0.0506 

9.9640 

0.1667 

0.1146 

0.3179 

0.2909 

0.5365 

0.5261 

0.9506 

0.9489 

21 

.9579 

.8264 

.0523 

.9664 

.1689 

.1173 

.3208 

.2942 

.5411 

.5309 

.9618 

.9603 

22 

.9593 

.8287 

.0541 

.9687 

.1711 

.1200 

.3238 

.2975 

.5458 

.5358 

.9734 

.9719 

23 

.9607 

.8310 

.0558 

.9711 

.1733 

.1226 

.3268 

.3008 

.5505 

.5407 

.9853 

.9839 

24 

.9622 

.8333 

.0576 

.9735 

.1755 

.1253 

.3298 

.3041 

.5553 

.5457 

.9975 

.9961 

25 

9.9636 

9.8356 

0.0593 

9.9760 

0.1777 

0.1280 

0.3328 

0.3075 

0.5601 

0.5507 

1.0100 

1.0087 

26 

.9651 

.8379 

.0611 

.9784 

.1799 

.1308 

.3359 

.3109 

.5649 

.5557 

.0228 

.0216 

27 

.9665 

.8402 

.0628 

.9808 

.1821 

.1335 

.3389 

.3143 

.5698 

.5608 

.0361 

.0350 

28 

.9680 

.8425 

.0646 

.9832 

.1844 

.1362 

.3420 

.3177 

.5748 

.5660 

.0497 

.0487 

29 

.9695 

.8448 

.0664 

.9856 

.1867 

.1389 

.3451 

.3211 

.5798 

.5712 

.0638 

.0628 

30 

9.9709 

9.8471 

0.0682 

9.9880 

0.1889 

0.1417 

0.3482 

0.3245 

0.5848 

0.5764 

1.0783 

1.0774 

31 

.9724 

.8494 

.0700 

.9904 

.1912 

.1444 

.3514 

.3280 

.5899 

.5817 

.0934 

.0925 

32 

.9739 

.8517 

.0718 

.9929 

.1935 

.1472 

.3545 

.3315 

.5951 

.5871 

.1089 

.1081 

33 

.9754 

.8540 

.0736 

.9953 

.1958 

.1499 

.3577 

.3350 

.6003 

.5925 

.1250 

.1242 

34 

.9769 

.8563 

.0754 

.9977 

.1981 

.1527 

.3609 

.3385 

.605^ 

.5979 

.1416 

.1409 

35 

9.9784 

9.8586 

0.0772 

0.0002 

0.2004 

0.1555 

0.3641 

0.3420 

0.6110 

0.6034 

1.1590 

1.1583 

36 

.9798 

.8609 

.0790 

.0026 

.2028 

.1582 

.3674 

.3456 

.6164 

.6090 

.1770 

.1764 

37 

.9813 

.8632 

.0809 

.0051 

.2051 

.1610 

.3706 

.3491 

.6218 

.6147 

.1958 

.1952 

38 

.9829 

.8655 

.0827 

.0075 

.2075 

'.1638 

.3739 

.3527 

.6273 

.6204 

.2154 

.2149 

39 

.9844 

.8678 

.0845 

.0100 

.2098 

.1667 

.3772 

.3563 

.6329 

.6261 

.2359 

.2354 

40 

9.9859 

9.8701 

0.0864 

0.0124 

0.2122 

0.1695 

0.3805 

0.3599 

0.6386 

0.6319 

1.2573 

1.2569 

41 

.9874 

.8724 

.0883 

.0149 

.2146 

.1723 

.3839 

.3636 

.6443 

.6378 

.2799 

.2795 

42 

.9889 

.8748 

.0901 

.0173 

.2170 

.1751 

.3873 

.3673 

.6501 

.6438 

.3037 

.3033 

43 

.9904 

.8771 

.0920 

.0198 

.2194 

.1780 

.3907 

.3710 

.6560 

.6498 

.3288 

.3285 

44 

.9920 

.8794 

.0939 

.0223 

.2218 

.1808 

.3941 

.3747 

.6619 

.6559 

.3554 

.3552 

45 

9.9935 

9.8817 

0.0958 

0.0248 

0.2243 

0.1837 

0.3975 

0.3784 

0.6679 

0.6621 

1.3837 

1.3835 

46 

.9951 

.8840 

.0976 

.0272 

.2267 

.1866 

.4010 

.3822 

.6740 

.6684 

.4140 

.4138 

47 

.9966 

.8863 

.0995 

.0297 

.2292 

.1895 

.4045 

.3859 

.6802 

.6747 

.4465 

.4463 

48 

.9982 

.8887 

.1015 

.0322 

.2316 

.1924 

.4080 

.3897 

.6865 

.6811 

4815 

.4814 

49 

.9998 

.8910 

.1034 

.0347 

.2341 

.1953 

.4115 

.3936 

.6928 

.6876 

.5196 

.5195 

50 

0.0013 

9.8933 

0.1053 

0.0372 

0.2366 

0.1982 

0.4151 

0.3974 

0.6993 

0.6942 

1.5613 

1.5612 

51 

.0029 

.8956 

.1072 

.0397 

.2391 

.2011 

.4187 

.4013 

.7058 

.7008 

.6074 

.6073 

52 

.0044 

.8980 

.1092 

.0422 

.2416 

.2040 

.4223 

.4052 

.7124 

.7076 

.6588 

.6587 

53 

.0060 

.9003 

.1111 

.0447 

.2442 

.2070 

.4260 

.4091 

.7191 

.7144 

.7171 

.7171 

54 

.0076 

.9026 

.1131 

.0473 

.2467 

.2099 

.4297 

.4130 

.7259 

.7214 

.7844 

.7843 

55 

0.0092 

9.9050 

0.1150 

0.0498 

0.2493 

0.2129 

0.4334 

0.4170 

0.7328 

0.7284 

1.8638 

1.8638 

56 

.0108 

.9073 

.1170 

.0523 

.2518 

.2159 

.4371 

.4210 

.7398 

.7355 

.9610 

.9610 

57 

.0124 

.9096 

.1190 

.0548 

.2544 

.2189 

.4408 

.4250 

.7469 

.7428 

2.0863 

2.0863 

58 

.0140 

.9120 

.1209 

.0574 

.2570 

.2219 

.4446 

.4291 

.7541 

.7501 

.2627 

.2627 

59 

.0156 

.9143 

.1229 

.0599 

.2596 

.2249 

.4485   .4331 

.7615 

.7576 

2.5640 

2.5640 

60 

0.0172 

99167 

0.1249 

0.0625 

0.2623 

0.2279 

0.4523  !  0.4372 

0.7689 

0.7652 

Inf.   7n/_J 

77 


TAKLE    II. 


LOGARITHMS  OF  NUMBERS. 

1 

Proportional  Parts, 

la 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

i* 

1 

2  3 

4 

§ 

0 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

4 

8  12 

17 

21 

25 

29 

33 

37 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

4 

8 

11 

15 

19 

23 

26 

30 

34 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

3 

7 

10 

14 

17 

21 

24 

28 

31 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

3 

6 

10 

13 

16 

19 

"21. 

& 

29 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

3 

6 

9 

12 

15 

18 

21 

SM 

27 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

3 

6 

8 

11 

14 

17 

20 

22 

25 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

3 

5 

S 

11 

13 

16 

18 

21 

24 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

2 

5 

7 

10 

12 

15 

17 

20 

22 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

2 

5 

7 

9 

12 

14 

16 

19 

21 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

2 

4 

7 

9 

11 

13 

16 

18 

20 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

2 

4 

6 

8 

11 

13 

15 

17 

19 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

2 

4 

6 

8 

10 

12 

14 

16 

18 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

2 

4 

6 

8 

10 

12 

14 

15 

17 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

2 

4 

6 

7 

9 

11 

13 

15 

17 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

2 

4 

5 

7 

9 

11 

12 

14 

16 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

2 

3 

5 

7 

9 

10 

12 

14 

15 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

2 

3 

5 

7 

8 

10 

11 

13 

15 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

2 

3 

5 

6 

8 

9 

11 

13 

14 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

2 

3 

5 

6 

8 

9 

11 

12 

14 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

1 

3 

4 

6 

7 

9 

10 

12 

13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

3 

4 

6 

7 

9 

10 

11 

13 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

3 

4 

6 

7 

8 

10 

11 

12 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

3 

4 

5 

7 

8 

9 

11 

12 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

3 

4 

5 

6 

8 

9 

10 

12 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

3 

4 

5 

6 

8 

9 

10 

11 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

2 

4 

5 

6 

7 

9 

10 

11 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

2 

4 

5 

6 

7 

8 

10 

11 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

577£ 

5786 

2 

3 

5 

6 

7 

8 

9 

10 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

2 

3 

5 

6 

7 

8 

9 

10 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

2 

3 

4 

5 

7 

8 

9 

10 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

2 

3 

4 

5 

6 

8 

9 

10 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

2 

3 

4 

5 

6 

7 

8 

9 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

2 

3 

4 

5 

G 

7 

8 

9 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

2 

3 

4 

5 

6 

7 

8 

9 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

2 

3 

4 

5 

6 

7 

8 

9 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

2 

3 

4 

5 

6 

7 

8 

9 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

2 

3 

4 

5 

6 

'7 

7 

8 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

2 

3 

4 

5 

5 

G 

7 

8 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

2 

3 

4 

4 

5 

6 

7 

8 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

2 

3 

4 

4 

5 

G 

7 

8 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

2 

3 

3 

4 

5 

G 

7 

8 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143  7152 

2 

3 

3 

4 

5 

6 

7 

8 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226  7235 

2j  2 

3 

4 

5 

6 

7 

7 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308  7316 

2  2 

3 

4 

5 

6 

6 

7 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380  j  7388  7396 

1 

2  2!  3 

4 

5 

6 

6j  7 

78 


TASLE    SI. 


LOGARITHMS  OF  NUMBERS. 

Natural 
Numbers. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  Parts, 

1 

2  3 

4 

i 

6 

7 

8 

9 

.55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

! 

2 

3 

.. 

5 

j 

6 

7 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

] 

2 

€ 

4 

5 

5 

6 

7 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

] 

f 

3 

4 

5 

5 

6 

7 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

1 

1 

3 

4 

.. 

| 

6 

7 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

] 

1 

3 

4 

4 

5 

6 

7 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

1 

3 

4 

5 

6 

6 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

1 

3 

4 

4 

u 

G 

6 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

1 

3 

« 

4 

5 

6 

6 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

1 

3 

3 

4 

5 

5 

6 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

1 

2 

3 

3 

4 

5 

5 

6 

65 

"8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

1 

2 

3 

3 

4 

5 

5 

6 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

1 

3 

* 

4 

5 

5 

6 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

1 

3 

4 

5 

5 

6 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

1 

3 

4 

4 

5 

6 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

1 

1 

2 

4 

4 

5 

6 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

1 

1 

2 

4 

4 

5 

6 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

1 

1 

2 

4 

i 

5 

5 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

1 

1 

2 

4 

4 

5 

5 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

1 

1 

2 

4 

4 

5 

5 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

1 

1 

2 

4 

4 

5 

5 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

1 

1 

2 

2 

3 

4 

5 

5 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

1 

1 

2 

2 

o 

4 

5 

5 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

1 

1 

2 

2 

8 

3 

4 

4 

5 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

896^ 

8971 

1 

1 

2 

2 

3 

8 

4 

4 

5 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

1 

1 

2 

2 

2 

3 

4 

4 

5 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

1 

2 

2 

3 

3 

4 

4 

5 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

1 

2 

2 

3 

3 

4 

4 

5 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

1 

2 

2 

3 

3 

4 

4 

5 

S3 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

1 

2 

2 

3 

3 

4 

4 

5 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

1 

2 

2 

8 

8 

4 

4 

5 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

1 

2 

2 

8 

3 

4 

4 

5 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

1 

2 

2 

3 

3 

4 

4 

5 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

0 

1 

2 

2 

3 

3 

4 

4 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

0 

1 

2 

2 

3 

8 

4 

4 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

0 

1 

2 

2 

3 

8 

4 

4 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

0 

1 

2 

2 

8 

3 

4 

4 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

0 

1 

2 

2 

3 

3 

4 

4 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

0 

1 

2 

2 

8 

3 

4 

4 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

0 

1 

2 

2 

3 

3 

4 

4 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

0 

1 

2 

2 

3 

8 

4 

4 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

0 

1 

2 

2 

3 

3 

4 

4 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

0 

1 

2 

2 

3 

3 

4 

4 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

0 

1 

2 

2 

8 

3 

4 

4 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

0 

1 

2 

2 

3 

3 

4 

4 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

99911 

9996 

0 

'  1  2 

— 

2 

3 

3 

3 

4 

79 


TABLE  III.    MEAN  REFRACTION. 

Barometer  30  inches.    Fahrenheit's  Thermometer  50°. 


Apparent 
Altitude. 

Mean 
Refraction. 

.pparent 
Altitude. 

Mean 
Refraction. 

Apparent 
Altitude. 

Mean 
Refraction. 

Apparent 
Altitude. 

Mean 
.Refraction. 

Apparent 
Altitude. 

Mean 
Refraction. 

0      ( 

/      ii 

0      1 

9  30 

;      n 
5  35.1 

O      1 

15     0 

i      // 
3  34.1 

o    / 
25     0 

i      ii 
2     4.4 

0      1 

42     0 

i     ii 
1     4.7 

0     0 

36  29.4 

35 

5  32.4 

10 

3  31.7 

10 

2     3.4 

20 

1     3.9 

1     0 

24  53.6 

40 

5  29.6 

20 

3  29.4 

20 

2     2.5 

40 

1     3.2 

2     0 

18  25.5 

45 

5  27.0 

30 

3  27.1 

30 

2     1.6 

43     0 

1     2.4 

3     0 

14  25.1 

50 

5  24.3 

40 

3  24.8 

40 

2     0.7 

20 

1     1.7 

4     0 

11  44.4 

55 

5  21.7 

50 

3  22.6 

50 

1  59.8 

40 

1     1.0 

5     0 

9  52.0 

10     0 

5   19.2 

16     0 

3  20.5 

26     0 

1  58.9 

44     0 

1     0.3 

5 

9  44.0 

5 

5   16.7 

10 

3  18.4 

10 

1  58.1 

20 

0  59.6 

10 

9  36.2 

10 

5   14.2 

20 

3  16.3 

20 

1  57.2 

40 

0  58.9 

15 

9  28.6 

15 

5  11.7 

30 

3  14.2 

30 

1   56.4 

45     0 

0  58.2 

20 

9  21.2 

20 

5     9.3 

40 

3  12.2 

40 

1   55.5 

20 

0  57.6 

25 

9  14.0 

25 

5     6.9 

50 

3  10.3 

50 

1  54.7 

40 

0  56.9 

5  30 

9     7.0 

10  30 

5     4.6 

17     0 

3     8.3 

27     0 

1  53.9 

46     0 

0  56.2 

35 

9     0.1 

35 

5     2.3 

10 

3     6.4 

10 

1  53.1 

20 

0  55.6 

40 

8  53.4 

40 

5     0.0 

20 

3     4.6 

20 

1   52.3 

40 

0  55.0 

45 

8  46.8 

45 

4  57.8 

30 

3     2.8 

30 

1  51.5 

47     0 

0  54.3 

50 

8  40.4 

50 

4  55.6 

40 

3     1.0 

40 

1  50.7 

20 

0  53.7 

55 

8  34.2 

55 

4  53.4 

50 

2  59.2 

50 

1  50.0 

40 

0  53.1 

6    0 

8  28.0' 

11     0 

4  51.2 

18     0 

2  57.5 

28     0 

1  49.2 

48     0 

0  52.5 

5 

8  22.1 

5 

4  49.1 

10 

2  55.8 

20 

1  47.7 

49     0 

0  50.6 

10 

8  16.2 

10 

4  47.0 

20 

2  54.1 

40 

1  46.2 

50     0 

0  48.9 

15 

8  10.5 

15 

4  44.9 

30 

2  52.4 

29     0 

1  44.8 

51     0 

0  47.2 

20 

8     4.8 

20 

4  42.9 

40 

2  50.8 

20 

1  43.4 

52     0 

0  45.5 

25 

7  59.3 

25 

4  40.9 

50 

2  49.2 

40 

1  42.0 

53     0 

0  43.9 

6  30 

7  53.9 

11  30 

4  38.9 

19     0 

2  47.7 

30     0 

1  40.6 

54     0 

0  42.3 

35 

7  48.7 

35 

4  36.9 

10 

2  46.1 

20 

1  39.3 

55     0 

0  40.8 

40 

7  43.5 

40 

4  35.0 

20 

2  44.6 

40 

1  38.0 

56     0 

0  39.3 

45 

7  38.4 

45 

4  33.1 

30 

2  43.1 

31     0 

1   36.7 

57     0 

0  37.8 

50 

7  33.5 

50 

4  31.2 

40 

2  41.6 

20 

1  35.5 

58     0 

0  36.4 

55 

7  28.6 

55 

4  29.4 

50 

2  40.2 

40 

1  34.2 

59     0 

0  35.0 

7    0 

7  23.8 

12     0 

4  27.5 

20     0 

2  38.8 

32    0 

1  33.0 

60     0 

0  33.6 

5 

7  19.2 

5 

4  25.7 

10 

2  37.4 

20 

1  31.8 

61     0 

0  32.3 

10 

7  14.6 

10 

4  23.9 

20 

2  36.0 

40 

1  30.7 

62     0 

0  31.0 

15 

7  10.1 

15 

4  22.2 

30 

2  34.6 

33     0 

1  29.5 

63     0 

0  29.7 

20 

7     5.7 

20 

4  20.4 

40 

2  33.3 

20 

1  28.4 

64     0 

0  28.4 

25 

7     1.4 

25 

4  18.7 

50 

2  32.0 

40 

1  27.3 

65     0 

0  27.2 

7  30 

6  57.1 

12  30 

4  17.0 

21     0 

2  30.7 

34     0 

1  26.2 

66     0 

0  25.9 

35 

6  53.0 

35 

4  15.3 

10 

2  29.4 

20 

1  25.1 

67     0 

0  24.7 

40 

6  48.9 

40 

4  13.6 

20 

2  28.1 

40 

1   24.1 

68     0 

0  23.6 

45 

6  44.9 

45 

4  12.0 

30 

2  26.9 

35     0 

1  23.1 

69     0 

0  22.4 

50 

6  41.0 

50 

4  10.4 

40 

2  25.7 

20 

1   22.0 

70     0 

0  21.2 

55 

6  37.1 

55 

4     8.8 

50 

2  24.5 

40 

1  21.0 

71     0 

0  20.1 

8     0 

6  33.3 

13     0 

4     7.2 

22     0 

2  23.3 

36     0 

1  20.1 

72     0 

0  18.9 

5 

6  29.6 

5 

4     5.6 

10 

2  22.1 

20 

1  19.1 

73     0 

0  17.8 

10 

6  25.9 

10        4     4.1 

20 

2  20.9 

40 

1   18.2 

74     0 

0  16.7 

15 

6  22.3 

15 

4     2.6 

30 

2  19.8 

37     0 

1   17.2 

75     0 

0  15.6 

20 

6  18.8 

20 

4     1.0 

40 

2  18.7 

20 

1   16.3 

76     0 

0  14.5 

25 

6  15.3 

25 

3  59.6 

50 

2  17.5 

40 

1  15.4 

77  '0 

0  13.5 

8  30 

6  11.9 

13  30 

3  58.1 

23     0 

2  16.4 

38     0 

1   14.5 

78    0 

0  12.4 

35 

6     8.5 

35 

3  56.6 

10 

2  15.4 

20 

1   13.6 

79     0 

0  11.3 

40 

6     5.2 

40 

3  55.2 

20 

2  14.3 

40 

1   12.7 

80     0 

0  10.3 

45 

6     2.0 

45 

3  53.7 

30 

2  13.3 

39     0 

1   11.9 

81     0 

0     9.2 

50 

5  58.8 

50 

3  52.3 

40 

2  12.2 

20 

1   11.0 

82     0 

0     8.2 

55 

5  55.7 

55 

3  50.9 

50 

2  11.2 

40 

1   10.2 

83     0 

0     7.2 

9     0 

5  52.6 

14     0 

3  49.5 

24     0 

2  10.2 

40     0 

1     9.4 

84     0 

0     6.1 

p 

5  49.6 

10 

3  46.8 

10 

2     9.2 

20 

1     8.6 

85     0 

0     5.1 

10 

5  46.6 

20 

3  44.2 

20 

2     8.2 

40 

1     7.8 

86     0 

0     4.1 

15 

5  43.6 

30 

3  41.6 

30 

2     7.2 

41     0 

1     7.0 

87     0 

0     3.1 

20 

5  40.7 

40 

3  39.0 

40 

2     6.2 

20 

1     6.2 

88     0 

0     2.0 

25 

5  37.9 

50 

3  36.5 

50 

2     5.3 

40 

1     5.4 

89     0 

0     1.0 

9  30 

5  35.1 

15     0 

3  34.1 

25     0 

2     4.4 

42     0 

1     4.7 

90     0 

0     0.0 

TABLE    III.  A. 

Correction  of  the  Mean  Refraction  for  the  Height  of  the  Barometer. 


Barometer. 

MEAN     REFRACTION. 

Barometer. 

O' 

1' 

2' 

3' 

4' 

5' 

6' 

•y, 

§' 

9' 

10' 

Subtract. 

"I 

( 

I 

II 

Add. 

0 

30 

0 

31) 

0 

30 

0 

3» 

0 

30 

6' 

a'i 

0 

:*o 

o  so' 

6' 

30 

6    3O 

6 

27.50 

51 

0 

2 

5 

7 

10 

12 

15 

17 

20 

23 

25 

28 

30 

33 

35  38 

40 

43 

45  48 

27.55 

0 

2 

5 

7 

10 

12 

15 

17 

20 

7! 

25 

27 

30 

32 

35  37 

40 

42 

45 

47 

50 

27.60 

0 

2 

5 

7 

10 

12 

14J17 

19 

24  27 

29 

31 

34  36 

39 

41 

44 

40 

49 

27.65 

0 

2 

5 

7 

9 

12 

14|l6 

19 

21 

24!  26 

28 

31 

33!  36 

88 

40 

4,3 

45 

48 

27.70 

0 

2 

5 

7 

9 

11 

14 

10 

18 

21 

23 

25 

28 

30 

32  35 

.37 

39 

42 

44 

47 

27.75 

0 

2 

4 

7 

9 

11 

13 

16 

18 

20 

23 

25 

27 

29 

32!34 

36 

39 

41 

43 

46 

27.80 

0 

2 

4 

7 

9 

11 

13  15 

18 

20 

22 

24 

27 

29 

31i33 

.35 

38 

4!) 

42 

45 

27.85 

0 

2 

4 

G 

9 

11 

13  15 

17 

19 

22 

24 

26 

28 

3032 

35 

.37 

39 

41 

44 

27.90 

0 

2 

.  4 

6 

8 

10 

13 

15 

17 

19 

21 

23 

25 

27 

30  32 

34 

.30 

38 

40 

43 

27.95 

0 

2 

4 

0 

8 

10 

12 

14 

16 

18 

21 

23 

25 

27 

.29 

31 

33 

35 

37 

39 

42 

28.00 

0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

.30 

32 

34 

30 

38 

41 

28.05 

o 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

'2'2 

24 

25 

27 

29 

31 

33 

,35 

37 

39 

28.10 

0 

2 

4 

6 

8 

9 

11 

13 

15 

17 

14 

21 

23 

25 

27 

29 

31 

.3.3 

34 

30 

38 

28.15 

0 

2 

4 

0 

7 

9 

11 

1.3 

15 

17 

19 

20 

22 

24 

546 

28 

30 

32 

34 

30 

37 

28.20 

0 

2 

4 

5 

7 

9 

11 

1,3 

14 

10 

18 

20 

22 

24 

25 

27 

29 

.31 

3,3 

35 

36 

28.25 

0 

2 

a 

5 

7 

9 

10 

12 

14 

16 

18 

19 

21 

2,3 

25 

20 

28 

30 

32 

34 

35 

28.30 

0 

2 

3 

5 

7 

8 

10 

12 

14 

15 

17 

19 

21 

22 

24 

20 

27 

29 

31 

33 

34 

28.35 

0 

2 

.3 

5 

7 

8 

10 

12 

1.3 

15 

17 

18 

20 

22 

2» 

25 

27 

28 

30 

32 

33 

28.40 

0 

2 

3 

5 

0 

8 

10 

11 

13 

14 

16 

18 

19 

21 

23J24 

20 

27 

29 

31 

32 

28.45 

0 

2 

9 

5 

6 

8 

9 

11 

12 

14 

10 

17 

19 

20 

22 

23 

25 

27 

28 

30 

31 

28.50 

0 

1 

3 

4 

6 

7 

9 

10 

12 

14 

15 

17 

18 

20 

21 

23 

24 

26 

27 

29 

30 

31.50 

28.55 

0 

1 

3 

4 

6 

7 

9 

10 

12 

13 

15 

10 

37 

19 

20 

22 

2.3 

25 

20 

28 

29 

31.45 

28.60 

0 

1 

9 

4 

6 

7 

8 

10 

11 

1.3 

14 

15 

17 

18 

20 

21 

23 

24 

25 

27 

28 

31.40 

28.65 

0 

9 

4 

5 

7 

8 

9 

11 

12 

14 

15 

16 

18 

19 

20 

22 

23 

25 

20 

27 

31.35 

28.70 

0 

3 

4 

5 

G 

8 

9 

10 

12 

13 

14 

16 

17 

18 

20 

21 

22 

24 

25 

26 

31.30 

28.75 

0 

2 

4 

5 

G 

7 

9 

10 

11 

13 

14 

15 

10 

18 

19 

20 

21 

23 

24 

25 

31.25 

28.80 

0 

2 

4 

5 

G 

7 

8 

10 

11 

12 

13 

14 

10 

17 

18 

19 

21 

22 

2.3 

24 

31.20 

2885 

0 

2 

8 

5 

0 

7 

8 

9 

10 

12 

1,3 

14 

15 

10 

17 

19 

20 

21 

22 

23 

31.15 

28.90 

0 

2 

3 

4 

9 

7 

8 

9 

10 

11 

12 

1,3 

14 

16 

17 

18 

19 

20 

21 

22 

31.10 

28.95 

0 

2 

8 

4 

5 

6 

7 

8 

9 

11 

12 

13 

14 

15 

10 

17 

18 

19 

20 

21 

31.05 

29.00 

0 

2 

,3 

4 

5 

6 

7 

8 

9 

10 

11 

Vt 

1,3 

14 

15 

10 

17 

18 

19 

20 

31.00 

29.05 

0 

2 

8 

4 

5 

6 

7 

8 

9 

10 

11 

11 

12 

13 

14 

15 

10 

17 

18 

19 

30.95 

29.10 

0 

2 

a 

4 

4 

5 

G 

7 

8 

9 

10 

11 

12 

1,3 

14 

15 

15 

16 

17 

18 

30.90 

29.15 

0 

2 

3 

3 

4 

5 

6 

7 

8 

9 

9 

10 

11 

12 

1.3 

14 

15 

15 

10 

17 

30.85 

29.20 

0 

2 

2 

3 

4 

5 

G 

6 

7 

8 

9 

10 

10 

11 

12 

1,3 

14 

15 

15 

16 

30.80 

29.25 

0 

2 

a 

4 

4 

5 

6 

7 

8 

8 

9 

10 

11 

11 

12 

13 

14 

14 

15 

30.75 

29.30 

0 

2 

3 

3 

4 

5 

6 

G 

7 

8 

8 

9 

10 

11 

11 

12 

13 

13 

14 

30-70 

29.35 

0 

2 

3 

a 

4 

5 

5 

G 

7 

7 

8 

9 

9 

10 

10 

11, 

12 

13 

13 

30.65 

29.40 

0 

1 

2 

2 

3 

4 

4 

5 

5 

G 

7 

7 

8 

8 

9 

10 

10 

11 

12 

12 

30.60 

29.45 

0 

1 

2 

2 

a 

3 

4 

4 

5 

G 

6 

7 

7 

8 

8 

9 

9 

10 

11 

11 

30.55 

29.50 

0 

0 

I» 

1 

2 

2 

3 

3 

4 

5 

5 

G 

G 

7 

7 

8 

8 

9 

9 

10 

10 

30.50 

29.55 

0 

0 

1 

1 

2 

2 

3 

3 

4 

4 

5 

5 

5 

6 

G 

7 

7 

8 

8 

9 

9 

30.45 

29.60 

0 

0 

1 

1 

2 

2 

2 

3 

3 

4 

4 

4 

5 

5 

G 

6 

G 

7 

7 

8 

8 

30.40 

29.65 

0 

0 

1 

1 

1 

2 

2 

2 

3 

a 

4 

4 

4 

5 

5 

5 

G 

G 

6 

7 

7 

30.35 

29.70 

0 

0 

1 

1 

1 

1 

2 

2 

2 

3 

3 

a 

4 

4 

4 

5 

5 

5 

5 

6 

6 

30.30 

29.75 

0 

0 

0 

1 

1 

1 

1 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

4 

5 

5 

5 

30.25 

29.80 

0 

0 

0 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

3 

3 

a 

8 

3 

4 

4 

4 

30.20 

29.85 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

2 

2 

2    2 

2 

2 

2 

3 

3 

3 

3 

30.15 

29.90 

0 

0 

0 

0 

0 

0 

1 

I 

1 

1 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

2 

30.10 

29.95 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

30.05 

30.00 

0 

0 

0 

0 

0 

0 

0 

0 

00 

0 

0 

0 

0    0 

0 

0 

0 

0 

0 

0 

0 

30.00 

Subtract, 

6' 

30' 

6' 

30 

6' 

30 

6' 

36" 

H 

0 

30* 

o 

ai 

6'    30 

6' 

30 

6' 

30 

6' 

3« 

6' 

Add. 

Barometer. 

0' 

I' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

10' 

MEAN      REFRACTION. 

11 


81 


TABLE    III.  B. 

Correction  of  the  Mean  Refraction  for  the  Heisht  of  the  Thermometer. 


Thermom. 

MEAN  REFRACTION. 

Thermom. 

0' 

1' 

2' 

3' 

4' 

5' 

6' 

»' 

8' 

9' 

10 

Add. 

0 

36 

<S 

30 

0 

3* 

\ 

3j 

0 

30 

0 

30* 

6' 

aft 

& 

3/> 

0 

;;ft 

o  36 

0 

Add. 

0 

—10 

i 

0 

4 

6 

II 
12 

ii 

90 

i 

24 

a 

28 

33 

87 

41 

J  „ 

-l8 

46 

5C 

55 

GO 

05 

70 

75 

80  85 

-90 

—  8 

0 

4 

s 

12 

15 

19 

28 

27 

31 

36 

40 

44 

48 

53 

58 

62 

07 

72 

77 

82 

87 

—  8 

—  6 

0 

4 

7 

11 

Ifi 

19 

22 

26 

30 

34 

38 

42 

47 

51 

55 

60 

04 

09 

74 

79 

84 

—  6 

-  4 

0 

4 

7 

11 

14 

18 

22 

25 

29 

33 

37 

41 

45 

49 

53 

57 

62 

00 

71 

70 

80 

—  4 

—  2 

o 

3 

7 

10 

14 

17 

21 

24 

28 

31 

35 

39 

43 

47 

51 

55 

59 

04 

OS 

72 

77 

.  .  o 

0 

Q 

3 

7 

10 

13 

10 

20 

23 

27 

30 

34 

37 

41 

45 

49 

53 

57 

01 

05 

69 

74 

0 

2 

0 

3 

6 

9 

12 

16 

19 

22 

25 

29 

32 

36 

39 

43 

47 

50 

54 

58 

62 

GO 

70 

2 

4 

0 

3 

G 

9 

12 

15 

18 

21 

24 

28 

31 

34 

37 

41 

44 

48 

52 

55 

59 

03 

67 

4 

6 

0 

3 

6 

B 

11 

H 

17 

20 

23 

20 

29 

32 

30 

39 

42 

40 

49 

53 

50 

00 

64 

6 

8 

0 

3 

5 

S 

11 

14 

10 

19 

22 

25 

28 

31 

34 

37 

40 

43 

47 

50 

54 

57 

61 

8 

10 

0 

3 

5 

8 

10 

13 

15 

18 

21 

24 

20 

29 

32 

35 

38 

41 

44 

48 

51 

54 

58 

10 

11 

0 

2 

5 

7 

10 

13 

15 

18 

20 

23 

20 

28 

31 

34 

37 

40 

43 

40 

49 

53 

56 

11 

.   12 

0 

2 

5 

7 

10 

12 

15 

17 

20 

22 

25 

28 

30 

33 

36 

39 

42 

45 

48 

51 

54 

12 

13 

0 

2 

5 

7 

9 

12 

14 

17 

19 

22 

24 

27 

30 

32 

35 

38 

41 

44 

47 

50 

53 

13 

14 

0 

2 

5 

7 

9 

11 

14 

10 

19 

21 

24 

26 

29 

31 

34 

37 

40 

42 

15 

48 

51 

14 

15 

0 

2 

4 

7 

9 

11 

13 

16 

18 

20 

|S 

25 

28 

30 

§3 

30 

38 

41 

44 

47 

50 

15 

16 

0 

2 

4 

6 

9 

11 

13 

15 

18 

20 

22 

25 

27 

29 

32 

35 

37 

40 

43 

45 

48 

16 

17 

0 

2 

4 

6 

8 

10 

It 

15 

17 

19 

21 

24 

20 

2;» 

31 

33 

36 

'39 

41 

44 

47 

17 

18 

0 

2 

4 

G 

8 

10 

12 

14 

16 

19 

21 

23 

25 

28 

80 

32 

35 

37 

40 

43 

45 

18 

19 

0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

27 

29 

31 

34 

36 

39 

41 

44 

19 

20 

0 

2 

4 

6 

8 

9 

11 

13 

15 

17 

19 

22 

24 

20 

28 

SO 

33 

35 

37 

40 

42 

20 

21 

0 

2 

4 

5 

7 

9 

11 

13 

15 

17 

19 

21 

23 

25 

27 

29 

31 

34 

30 

38 

41 

21 

22 

0 

2 

3 

5 

7 

9 

11 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

35 

37 

39 

22 

23 

0 

2 

3 

5 

7 

8 

10 

12 

14 

15 

17 

19 

21 

23 

25 

27 

29 

31 

33 

30 

38 

23 

24 

0 

2 

3 

5 

G 

8 

10 

11 

13 

15 

17 

18 

20 

22 

24 

20 

28 

30 

32 

34 

36 

24 

25 

0 

2 

3 

5 

G 

8 

g 

11 

13 

14 

1C 

18 

19 

21 

23 

25 

27 

29 

31 

33 

35 

25 

26 

0 

3 

4 

6 

7 

9 

11 

12 

14 

15 

17 

19 

20 

22 

24 

20 

28 

29 

31 

33 

26 

27 

0 

3 

4 

6 

7 

9 

10 

12 

13 

15 

10 

18 

19 

21 

23 

25 

20 

38 

30 

32 

27 

28 

0 

3 

4 

5 

7 

8 

10 

11 

12 

14 

15 

17 

19 

20 

22 

23 

25 

27 

29 

30 

28 

29 

0 

3 

4 

5 

6 

8 

9 

11 

12 

13 

15 

16 

18 

19 

21 

22 

24 

20 

27 

29 

29 

30 

0 

2 

4 

5 

6 

7 

9 

10 

11 

13 

14 

15 

17 

IS 

20 

21 

23 

24 

20 

28 

30 

31 

Q 

2 

3 

5 

G 

7 

8 

9 

11 

12 

13 

15 

10 

17 

19 

20 

22 

23 

25 

26 

31 

32 

0 

1 

2 

3 

4 

6 

7 

8 

9 

10 

11 

13 

14 

15 

16 

18 

19 

20 

22 

23 

25 

32 

33 

0 

1 

2 

3 

4 

5 

G 

7 

8 

10 

11 

12 

13 

14 

15 

17 

18 

19 

21 

22 

23 

33 

34 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

10 

11 

12 

13 

14 

16 

17 

18 

19 

21 

22 

34 

35 

0 

2 

3 

4 

5 

6 

G 

7 

8 

9 

10 

11 

13 

14 

15 

10 

17 

18 

19 

20 

35 

36 

0 

2 

3 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

10 

17 

18 

19 

36 

37 

0 

2 

2 

3 

4 

5 

6 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

37 

38 

0 

2 

3 

4 

4 

5 

6 

7 

7 

8 

9 

10 

11 

12 

13 

13 

14 

15 

16 

38  ' 

39 

0 

2 

3 

a 

4 

5 

5 

6 

7 

8 

8 

9 

10 

11 

11 

12 

13 

14 

15 

39 

40 

0 

2 

2 

a 

4 

4 

5 

G 

6 

7 

8 

S 

9 

10 

10 

11 

12 

13 

13 

40 

41 

0 

2 

2 

3 

3 

4 

4 

5 

6 

G 

7 

7 

8 

9 

9 

10 

11 

11 

12 

41 

42 

0 

0 

1 

2 

2 

3 

3 

4 

4 

5 

5 

6 

7 

7 

8 

8 

9 

9 

10 

11 

42 

43 

0 

0 

1 

2 

2 

3 

3 

3 

4 

4 

5 

5 

6 

6 

7 

7 

8 

8 

9 

9 

43 

44 

0 

0 

1 

1 

2 

2 

a 

3 

3 

4 

4 

4 

5 

5 

6 

6 

7 

7 

8 

8 

44 

45 

0 

0 

1 

1 

1 

1 

2 

2 

2 

a 

3 

3 

4 

4 

4 

5 

5 

0 

G 

6 

7 

45 

46 

0 

0 

0 

1 

1 

1 

1 

2 

2 

2 

2 

2 

3 

3 

4 

4 

4 

4 

5 

5 

5 

46 

47 

0 

0 

0 

1 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

47 

48 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

2 

2 

3 

48 

49 

0 

0 

0 

0 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

49 

50 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

50 

Add. 

<s 

30 

0 

3ft 

II 
0 

3ft 

0 

3ft 

61 

3ft 

6' 

3ft 

6' 

3ft 

£ 

30 

6  3# 

o' 

30 

0 

Add. 

i 

| 

Xftcrinom. 

O' 

1' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

»' 

10' 

Tlicrnioro* 

MEAN  REFRACTION. 

TABLE    III.  B. 

Corrector  of  the  Mean  Refraction  for  the  Height  of  the  Thermometer. 


Thermom. 

MEAN      REFRACTION. 

Thermom. 

0 

1' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

10' 

Subtract.  ' 

1 

30 

d 

s£ 

6 

9& 

0 

3<'l 

0 

3& 

II 

0 

30 

0 

3l 

o' 

:w 

6' 

3& 

6'  36 

6' 

Subtract. 

o 

( 

50 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

6'j  6 

c 

50° 

51 

i) 

0 

0 

0 

0 

0 

0 

0 

0 

1 

l 

1 

1 

i 

1 

1 

1 

1 

ij  i 

i 

51 

52 

0 

0 

0 

0 

0 

1 

I 

1 

] 

1 

1 

1 

1 

2 

2 

2 

2 

2 

2!   2 

3 

52 

53 

0 

0 

0 

1 

i 

1 

1 

1 

I 

2 

2 

2 

2 

2 

2 

a 

3 

3 

3 

4 

4 

53 

54 

0 

0 

0 

1 

i 

1 

1 

2 

2 

2 

2 

3 

3 

a 

8 

4 

4 

4 

5 

5 

5 

54 

55 

0 

0 

1 

1 

i 

1 

2 

2 

2 

3 

3 

3 

4 

4 

4 

5 

5 

', 

6 

G 

6 

55 

.56 

0 

0 

1 

1 

i 

2 

2 

2 

3 

3 

4 

4 

4 

5 

5 

6 

G 

G 

7 

7 

8 

56 

57 

0 

0 

1 

1 

2 

2 

2 

3 

a 

4 

4 

5 

5 

G 

6 

G 

7 

8 

8 

8 

9 

57 

58 

0 

0 

1 

1 

2 

2 

3 

3 

4 

4 

5 

5 

6 

G 

7 

7 

8 

9 

9 

10 

10 

58 

59 

0 

1 

1 

2 

2 

3 

3 

4 

4 

5 

5 

6 

6 

7 

8 

8 

9 

10 

10 

11 

12 

59 

60 

0 

1 

2 

2 

3 

3 

4 

5 

5 

6 

7 

7 

8 

9 

9 

10 

11 

11 

12 

13 

60 

61 

.0 

1 

2 

3 

3 

4 

4 

5 

6 

7 

7 

8 

9 

9 

10 

1  ! 

12 

12 

13 

14 

61 

62 

0 

1 

2 

3 

3 

4 

5 

G 

6 

7 

8 

9 

9 

10 

11 

12 

13 

14 

15 

15 

62 

63 

0 

1 

2 

3 

4 

5 

5 

6 

7 

8 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

63 

64 

0 

2 

2 

3 

4 

5 

6 

7 

7 

8 

9 

10 

11 

12 

13 

14 

15 

1G 

17 

18 

64 

65 

0 

2 

3 

3 

4 

5 

G 

7 

8 

9 

10 

11 

12 

13 

14 

15 

1G 

17 

18 

19 

65 

66 

0 

2 

3 

4 

B 

6 

C) 

7 

8 

9 

10 

11 

12 

14 

15 

1G 

17 

18 

19 

20 

66 

67 

0 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

16 

17 

18 

19 

20 

22 

67 

68 

0 

2 

3 

4 

5 

G 

7 

8 

9 

11 

11 

13 

14 

15 

16 

18 

19 

20 

22 

23 

68 

69 

0 

2 

3 

4 

r, 

7 

8 

9 

10 

11 

12 

13 

15 

1C, 

17 

19 

20 

21 

23 

24 

69 

70 

0 

2 

3 

B 

6 

7 

8 

9 

10 

12 

12 

14 

1G 

17 

18 

20 

21 

22 

24 

25 

70 

71 

0 

2 

4 

5 

r, 

7 

8 

10 

11 

12 

13 

15 

1G 

IS 

19 

20 

22 

23 

25 

27 

71 

72 

0 

2 

4 

5 

G 

8 

9 

id 

11 

13 

14 

16 

17 

18 

20 

21 

23 

25 

26 

28 

72 

73 

0 

1 

3 

4 

5 

7 

8 

9 

11 

12 

13 

14 

16 

IS 

19 

21 

22 

24 

2G 

27 

29 

73 

74 

0 

1 

3 

4 

5 

7 

8 

10 

11 

12 

14 

15 

17 

18 

20 

22 

2,". 

25 

27 

28 

30 

74 

75 

0 

1 

3 

4 

6 

7 

8 

10 

11 

13 

14 

16 

IS 

19 

21 

22 

24 

26 

28 

29 

31 

75 

76 

0 

1 

3 

4 

8 

7 

9 

10 

12 

13 

15 

16 

18 

20 

22 

23 

25 

27 

29 

31 

32 

76 

77 

0 

1 

3 

5 

6 

8 

9 

11 

12 

14 

16 

17 

19 

21 

22 

24 

26 

28 

30 

32 

34 

77 

78 

0 

2 

3 

5 

r, 

8 

9 

11 

13 

14 

16 

18 

20 

21 

23 

25 

27 

29 

31 

33 

35 

78 

79 

0 

2 

3 

5 

6 

8 

10 

11 

13 

15 

17 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

79 

80 

0 

2 

3 

5 

7 

8 

10 

12 

14 

15 

17 

19 

21 

23 

25 

27 

29 

31 

83 

85 

37 

80 

81 

0 

2 

3 

5 

7 

9 

10 

12 

14 

16 

18 

20 

21 

24 

2G 

28 

30 

32 

34 

36 

38 

81 

82 

0 

2 

4 

5 

7 

9 

11 

13 

14 

16 

18 

20 

22  24 

2G 

28 

31 

33 

35 

37 

40 

82 

83 

0 

2 

4 

5 

7 

9 

11 

13 

15 

17 

19 

21 

23125 

27 

29 

31 

34 

36 

88 

41 

83 

84 

0 

2 

4 

6 

8 

9 

11 

13 

15 

17 

19 

21 

23  26 

1 

28 

30 

32 

35 

37 

39 

42 

84 

85 

0 

2 

4 

6 

8 

10 

12 

14 

1G 

IS 

20 

22 

24  26 

29 

31 

33 

3G 

33 

40 

43 

85 

86 

0 

2 

4 

6 

•8 

10 

12 

14 

16 

IS 

20 

28 

25  27 

29 

32 

34 

37 

39 

42 

44 

86 

87- 

0 

2 

4 

6 

8 

10 

12 

14 

17 

19 

21 

23 

25  28 

30 

32 

35 

38 

40 

43 

45 

87 

88 

0 

2 

4 

f, 

8 

10 

13 

15 

17 

19 

21 

24 

26;28 

31 

33 

36 

38 

41 

44 

46 

88 

89 

0 

2 

4 

6 

9 

11 

13 

15 

17 

20 

22 

24 

27  29 

32 

34 

37 

39 

4-2 

45 

48 

89 

90 

0 

2 

4 

7 

9 

11 

13 

1G 

18 

20 

23 

25 

27!30 

32 

35 

38 

40 

43 

46 

49 

90 

91 

0 

2 

4 

7 

9 

11 

14 

1G 

18 

21 

23 

25 

28  31 

33  36 

39 

41 

H 

47 

50 

91 

92 

0 

2 

5 

7 

9 

11 

14 

16 

19 

21 

24 

26 

29  31 

3437 

39 

42 

45 

48 

51 

92 

93 

0 

2 

5 

7 

9 

12 

14 

17 

19 

22 

24 

27 

29  32 

35  37 

40 

43 

4C, 

49 

52 

93 

94 

0 

2 

5 

7 

10 

12 

14 

17 

19 

22 

25 

27 

30  33 

35J38 

41 

44 

47 

50 

53 

94 

95 

0 

2 

5 

7 

10 

12 

15 

17 

20 

22 

25 

28 

30  33 

3G!39 

42 

45 

48 

51 

54 

95 

96 

0 

2 

5 

7 

10 

12 

IS 

18 

20 

23 

2G 

28 

31  34 

37  40 

'43 

4G 

49  52 

55 

96 

97 

0 

3 

5 

8 

10 

13 

15 

18 

21 

23 

2G 

29 

32  35 

38  41 

44  47 

50  j  53 

56 

97 

98 

0 

3 

5 

8 

10 

13 

16 

18 

21 

24 

27 

29 

32  35 

38141 

44  48 

51  54 

58 

98 

99 

0 

3 

5 

8 

11 

13 

16 

19 

21 

24 

27 

30 

33  36 

39 

42 

45  49 

52 

55 

59 

99 

100 

0 

3 

5 

8 

11 

13 

16 

19 

22 

25 

23 

31 

34  37 

40 

43 

46  50 

53 

56 

60 

100 

Subtract. 

4 

30 

d 

3('i 

0 

30 

0 

30 

6' 

3/i 

o' 

30 

0    3(J 

6' 

30 

6'   30 

n 

. 

30 

6 

Subtract. 

o 

1' 

2' 

3' 

4' 

5' 

6' 

V 

8' 

9' 

1O' 

HPViai-YV*rtv« 

Tnerrnom. 

MEAN      REFRACTION. 

i  nerinoni 

TABLE  IF.   SIDEREAL  INTO  MEAN  SOLAR  TIME. 


Sidd- 
real. 

oh 

lh' 

2h. 

3* 

41, 

5h. 

6h. 

7h. 

For 

^Arnnrta 

m. 

m.  a. 

m.  •. 

m.  B. 

m.  s. 

m.  • 

m.  g. 

m.  s. 

u.  • 

OUCOQU3. 

0 

0  00.000 

0  09.830 

0  19.659 

0  29.489 

0  39.318 

0  49.148 

0  58.977 

08.807 

g 

1 

0  00.164 

0  09.993 

0  19.823 

0  29.653 

0  39.482 

0  49.312 

0  59.141 

08.971 

1 

oioos 

2 

0  00.323 

0  10.157 

0  19.987 

0  29.816 

0  39.646 

0  49.475 

0  59.305 

09.185 

2 

.005 

3 

0  00.491 

0  10.321 

0  20.151 

0  29.980 

0  39.810 

0  49.639 

0  59.4G9 

09.298 

3 

.008 

4 

0  00.655 

0  10.485 

0  20.314 

0  30.144 

0  39.974 

0  49.803 

0  59.633 

09.462 

4 

.Oil 

5 

0  00.819 

0  10.649 

0  20.478 

0  30.308 

0  40.137 

0  49.967 

0  59.796 

09.626 

5 

.014 

6 

0  00.983 

0  10.813 

0  20.642 

0  30.472 

0  40.301 

0  50.131 

0  59.960J   09.790 

<3 

.016 

7 

0  01.147 

0  10.976 

0  20.806 

0  30.635 

0  40.405 

0  5U.2U5 

1  00.124 

09.954 

7 

.019 

8 

0  01.311 

0  11.140 

0  20.970 

0  30.799 

0  40.629 

0  5U.458 

1  00.288 

10.118 

8 

.022 

9 

0  01.474 

0  11.304 

0  21.134 

0  30.963 

0  40.793 

0  50.022 

1  00.452 

10.281 

c 

.025 

10 

0  01.638 

0  11.468 

0  21.297 

0  31.127 

0  40.956 

0  50.786 

1  00.616 

10.445 

14 

.027 

11 

0  01.802 

0  11.632 

0  21.461 

0  31.291 

0  41.120 

0  50.950  i  1  00.779 

10.609 

11 

.030 

12 

0  01.966 

0  11.795 

0  21.625 

0  31.455 

0  41.284 

0  51.114 

1  00.943 

10.773 

12 

.033 

13 

0  02.130 

0  11.959 

0  21.789 

0  31.618 

0  41.448 

0  51.278 

01.107 

10.937 

13 

.035 

14 

0^2.294 

0  12.123 

0  21.953 

0  31.782 

0  41.612 

0  51.441 

1  01.271 

11.100 

14 

.038 

15 

0  02.457 

0  12.287 

0  22.117 

0  31.946 

0  41.776 

0  51.605 

1  01.435 

11.264 

15 

.041 

16 

0  02.621 

0  12.451 

0  22.280 

0  32.110 

0  41.939 

0  51.769 

1  01.599 

11.428 

16 

.044 

17 

0  02.785 

0  12.615 

0  22.444 

0  32.274 

0  42.103 

0  51.933J  1  01.762 

11.592jj  17 

.040 

18 

0  02.949 

0  12.778 

0  22.608 

0  32.438 

0  42.267 

0  52.U97  1  01.926 

11.756 

18 

.049 

19 

0  03.113 

0  12.942 

0  22.772 

0  32.601 

0  42.431 

0  52.260 

1  02.090 

11.920 

19 

.052 

20 

0  03.277 

0  13.106 

0  22.936 

0  32.765 

0  42.595 

0  52.424 

1  02.254 

12.083 

20 

.055 

21 

0  03.440 

0  13.270 

0  23.099 

0  32.929 

0  42.759 

0  52.588 

1  02.418 

12.247 

21 

.057 

22 

0  03.604 

0  13.434 

0  23.263 

0  33.093 

0  42.922 

0  52.752  1  02.582 

12.411 

22 

.060  i 

23 

0  03.768 

0  13.598 

0  23.427 

0  33.257 

0  43.086 

0  52.916  1  02.745 

12.fi  75  !|  23 

.063 

24 

0  03.932 

0  13.761 

0  23.591 

0  33.420 

0  43.250 

0  53.080  1  02.909 

1  2.739  J  24 

.066 

25 

0  04.096 

0  13.925 

0  23.755 

0  33.584 

0  43.414 

0  53.243i  1  03.073 

12.G03JJ  25 

.068 

26 

0  04.259 

0  14.089 

0  23.919 

0  33.748 

0  43.578 

0  53.407 

1  03.237 

13.066  |  26 

.071 

27 

0  04.423 

0  14.253 

0  24.082 

0  33.912 

0  43.742 

0  53.571 

1  03.401 

13.230 

27 

.074 

28 

0  04.587 

0  14.417 

0  24.246 

0  34.076 

0  43.905 

0  53.735 

1  03.564 

13.394 

28 

.076  ! 

29 

0  04.751 

0  14.581 

0  24.410 

0  34.240 

0  44.069 

0  53.899 

1  03.728 

13.558 

29 

.079 

30 

0  04.915 

0  14.744 

0  24.574 

0  34.403 

0  44.233 

0  54.063 

1  03.892 

13.722 

30 

.082 

31 

0  05.079 

0  14.908 

0  24.738 

0  34.567 

0  44.397 

0  54.226 

1  04.056 

13.886 

31 

.085 

32 

0  05.242 

0  15.072 

0  24.902 

0  34.731 

0  44.561 

0  54.390 

1  04.220 

14.049 

32 

.087 

33 

0  05.406 

0  15.236 

0  25.065 

0  34.895 

0  44.724 

0  54  554 

04.384 

14.213 

33 

.090 

34 

0  05.570 

0  15.400 

0  25.229 

0  35.059 

0  44.888 

0  54.718 

04.547 

14.377 

34 

.093 

35 

0  05.734 

0  15.563 

0  25.393 

0  35.223 

0  45.052 

0  54.882 

04.711 

14.541 

35 

.096 

36 

0  05.898 

0  15.727 

0  25.557 

0  35.386 

0  45.216 

0  55.046 

04.875 

14.705 

36 

.098 

37 

0  06.062 

0  15.891 

0  25.721 

0  35.550 

0  45.380 

0  55.209 

05.039 

14.868 

37 

.101 

38 

0  06.225 

0  16.055 

0  25.885 

0  35.714 

0  45.544 

0  55.373 

05.203 

15.032 

38 

.104 

39 

0  06.389 

0  16.219 

0  26.048 

0  35.878 

0  45.707 

0  55.537 

05.367 

15.196 

39 

.106 

40 

0  06.553 

0  16.383 

0  26.212 

0  36.042 

0  45.871 

0  55.701 

05.530 

1  15.360 

40 

.109 

41 

0  06.717 

0  16.546 

0  26.376 

0  36.206 

0  46.035 

0  55.865 

05.694 

1  15.524 

41 

.112 

42 

0  06.881 

0  16.710 

0  26.540 

0  36.369 

0  46.199 

0  56.028 

1  05.858 

1  15.088 

42 

.115 

43 

0  07.045 

0  16.874 

0  26.704 

0  36.533 

0  46.363 

0  56.192 

1  06.022 

1  15.851 

43 

.117 

44 

0  07.208 

0  17.038 

0  26.867 

0  36.697 

0  46.527 

0  56.356 

1  06.186 

1  16.015 

44 

.120 

45 

0  07.372 

0  17.202 

0  27.031 

0  36.861 

0  46.690 

0  56.520 

1  06.350 

16.179 

45 

.123 

46 

0  07.536 

0  17.366 

0  27.195 

0  37.025 

0  46.854 

0  56.684 

1  06.513 

16.343 

46 

.126 

47 

0  07.700 

0  17.529 

0  27.359 

0  37.188 

0  47.018 

0  56.848 

1  06.677 

16.507 

47 

.128 

48 

0  07.864 

0  17.693 

0  27.523 

0  37.352 

0  47.182 

0  57.011 

1  08.841 

16.671 

48 

.131 

49 

0  08.027 

0  17.857 

0  27.687 

0  37.516 

0  47.346 

0  57.175 

1  07.005 

16.834 

49 

.134 

50 

0  08.191 

0  18.021 

0  27.850 

0  37.680 

0  47.510 

0  57.339 

1  07.160 

16.998 

50 

.137 

51 

0  08.355 

0  18.185 

0  28.014 

0  37.844 

0  47.673 

0  57.503 

1  07.332 

17.162 

51 

.139 

52 

0  08.519 

0  lb.349 

0  28.178 

0  38.008 

0  47.837 

0  57.667 

1  07.496 

1  7.326 

52 

.142 

53 

0  08.683 

0  18.512 

0  28.342 

0  38.171 

0  48.001 

0  57.831 

1  07.660 

17.490 

53 

.145 

54 

0  08.847 

0  18.676 

0  28.506 

0  38.335 

0  48.165 

0  57.994 

1  07.824 

17.654 

54 

.147 

55 

0  09.010 

0  18.840 

0  28.670 

0  38.499 

0  48.329 

0  58.158 

1  07.988 

1  17.817 

55 

.150 

56 

0  09.174 

0  19.004 

0  28.833 

0  38.663 

0  48.492 

0  58.322 

I  08.152 

1  17.981 

56 

.153 

57 

0  09.338 

0  19.168 

0  28.997 

0  38.827 

0  48.656 

0  58.486 

1  08.315 

1  18.145 

57 

.156 

58 

0  09.502 

0  19.331 

0  29.161 

0  38.991 

0  48.820 

0  58.650 

1  08.479 

I  18.309 

58 

.158 

59 

0  09.666 

0  19.495 

0  29.325 

0  39.154 

0  48.984 

0  58.814 

1  08.643 

1  18.473 

59 

.161  |] 

TABLE  IV.    SIDEREAL  INTO  MEAN  SOLAR  TIME. 


Side 
real 

8"- 

9h. 

10  "• 

ll* 

12  h- 

13  L 

14*. 

15s- 

For 

Soconds. 

m 

m.  t. 

m.  s. 

m.  i. 

in.  •< 

m.  i. 

m.  s. 

m.  8. 

m.  • 

o 

1  18.636 

1  28.466 

38.296 

1  48,125 

57.955 

2  07.784 

2  17.61 

2  27.443 

8 

i. 

i 

1  18.800 

1  28.630 

38.459 

1  48.289 

58.119 

2  07.948 

2  17.77 

2  27.607 

1 

0.003 

t 

1  18.964 

1  28.794 

38.623 

1  48.453 

58.282 

2  08.112 

2  17.94 

2  27.771 

j 

.005 

<• 

1  19.128 

1  28.958 

38.787 

1  48.617 

58.446 

2  08.276 

2  18.10 

2  27.935 

.008 

4 

1  19.292 

1  29.121 

38.951 

1  48.780 

58.610 

2  08.440 

2  18.26 

2  28.099 

4 

.011 

K 

I 

1  19.456 

1  29.285 

39.115 

1  48.944 

58.774 

2  08.603 

2  18.433 

2  28.263 

r 

.014 

6 

1  19.619 

1  29.449 

39.279 

I  49.108 

58.938 

2  08.767 

2  18.59 

2  28.426 

i 

.016 

j 

1  19.783 

1  29.613 

39.442 

1  49.272 

59.101 

2  08.931 

2  18.76 

2  28.590 

- 

.019 

8 

1  19.947 

1  29.777 

39.606 

1  49.436 

59.265 

2  09.095 

2  18.924 

2  28.754 

i 

.022 

fj 

1  20.111 

1  29.940 

39.77? 

1  49.600 

59.429 

2  09.259 

2  19.088 

2  28.918 

j 

.025 

10 

1  20.275 

1  30.104 

1  39.934 

1  49.763 

59.593 

2  09.423 

2  19.252 

2  29.082 

10 

.027 

11 

1  20.439 

I  30.268 

1  40.098 

1  49.927 

59.757 

2  09.586 

2  19.416 

2  29.245 

11 

.030 

12 

1  20.602 

1  30.432 

1  40.261 

1  50.091 

59.921 

2  09.750 

2  19.580 

2  29.409 

12 

.033 

13 

1  20.766 

1  30.596 

1  40.425 

1  50.255 

2  00.084 

2  09.914 

2  19.744 

2  29.573 

13 

.035 

14 

1  20.930 

1  30.760 

1  40.589 

1  50.419 

2  00.248 

2  10.078 

2  19.907 

2  29.737 

14 

.038 

15 

1  21.094 

1  30.923 

40.753 

1  50.583 

2  00.412 

2  10.242 

2  20.071 

2  29.901 

15 

.041 

16 

1  21.258 

1  31.087 

40.917 

1  50.746 

2  00.576 

2  10.405 

2  20.235 

2  30.065 

16 

.044 

17 

1  21.422 

1  31.251 

41.081 

1  50.910 

2  00.740 

2  10.569 

2  20.399 

2  30.228 

17 

.046 

18 

1  21.585 

1  31.415 

41.244 

1  51.074 

2  00.904 

2  10.733 

2  20.563 

2  30.392 

18 

.049 

19 

1  21.749 

1  31.579 

41.408 

1  51.238 

2  01.067 

2  10.897 

2  20.727 

2  30.556 

19 

,052 

20 

1  21.913 

1  31.743 

41.572 

1  51.402 

2  01.231 

2  11.061 

2  20.890 

2  30.720 

20 

.055 

21 

1  22.077 

1  31.906 

41.736 

1  51.565 

2  01.395 

2  11.225 

2  21.054 

2  30.884 

21 

.057 

22 

1  22.241 

1  32.070 

41.900 

1  51.729 

2  01.559 

2  11.388 

2  21.218 

2  31.048 

22 

.060 

23 

1  22.404 

1  32.234 

42.064 

1  51.893 

2  01.723 

2  11.552 

2  21.382 

2  31.211 

23 

.063 

24 

1  22.568 

1  32.398 

42.227 

1  52.057 

2  01.887 

2  11.716 

2  21.546 

2  31.375 

24 

.066 

25 

1  22.732 

1  32.562 

42.391 

1  52.221 

2  02.050 

2  11.880 

2  21.709 

2  31.539 

25 

.068 

26 

1  22.896 

1  32.726 

42.555 

1  52.385 

2  02.214 

2  12.044 

2  21.873 

2  31.703 

26 

.071 

27 

1  23.060 

1  32.889 

42.719 

1  52.548 

2  02.378 

2  12.208 

2  22.037 

2  31.867 

27 

.074 

28 

1  23.224 

1  33.053 

42.883 

1  52.712 

2  02.542 

2  12.371 

2  22.201 

2  32.031 

28 

.076 

29 

1  23.387 

1  33.217 

43.047 

1  52.876 

2  02.706 

2  12.535 

2  22.365 

2  32.194 

29 

.079 

30 

1  23.551 

1  33.381 

43.210 

1  53.040 

2  02.869 

2  12.699 

2  22.529 

2  32.358 

30 

.082 

31 

1  23.715 

1  33.545 

43.374 

1  53.204 

2  03.033 

2  12.863 

2  22.692 

2  32.522 

31 

.085 

32 

1  23.879 

1  33.708 

43.538 

1  53.368 

2  03.197 

2  13.027 

2  22.856 

2  32.686 

32 

.087 

33 

1  24.043 

1  33.872 

43.702 

1  53.531 

2  03.361 

2  13.191 

2  23.020 

2  32.850 

33 

.090 

34 

1  24.207 

1  34.036 

43.866 

1  53.695 

2  03.523 

2  13.354 

2  23.184 

2  33.013 

34 

.093 

35 

1  24.370 

1  34.200 

44.029 

1  53.859 

2  03.689 

2  13.518 

2  23.348 

2  33.177 

35 

.096 

36 

1  24.534 

1  34.364 

44.193 

1  54.023 

2  03.852 

2  13.682 

2  23.512 

2  33.341 

36 

.098 

37 

1  24.698 

1  34.528 

44.357 

1  54.187 

2  04.016 

2  13.846 

2  23.675 

2  33.505 

37 

.101 

38 

1  24.862 

1  34.691 

44.521 

1  54.351 

2  04.180 

2  14.010 

2  23.839 

2  33.669 

38 

.104 

39 

1  25.026 

1  34.855 

44.685 

1  54.514 

2  04.344 

2  14.173 

2  24.003 

2  33.833 

39 

JOG  i 

40 

1  25.190 

1  35.019 

44.849 

1  54.678 

2  04.508 

2  14.337 

2  24.167 

2  33.996 

40 

.109 

41 

1  25.353 

1  35.183 

45.012 

1  54.842 

2  04.672 

2  14.501 

2  24.331 

2  34.160 

41 

.112 

42 

1  25.517 

1  35.347 

45.176 

1  55.006 

2  04.835 

2  14.665 

2  24.495 

2  34.324 

42 

.115 

43 

1  25.681 

1  35.511 

45.340 

1  55.170 

2  04.999 

2  14.829 

2  24.658 

2  34.488 

43 

.117 

44 

1  25.845 

1  35.674 

45.504 

1  55.333 

2  05.163 

2  14.993 

2  24.822 

2  34.652 

44 

.120 

45 

1  26.009 

1  35.838 

45.668 

1  55.497 

2  05.327 

2  15.156 

2  24.986 

2  34.816 

45 

.123 

46 

1  26.172 

36.002 

45.832 

1  55.661 

2  05.491 

2  15.320 

2  25.150 

2  34.979 

46 

.126 

47 

1  26.336 

36.166 

1  45.995 

1  55.825 

2  05655 

2  15.484 

2  25.314 

2  35.143 

47 

.128 

48 

1  26.500 

36.330 

1  46.159 

1  55.989 

2  05.818 

2  15.b48 

2  25.477 

2  35.307 

48 

.131 

49 

1  26.664 

36.493 

1  46.323 

1  56.153 

2  05.982 

2  15.812 

2  25.641 

2  35.471 

49 

.134 

50 

1  26.828 

36.657 

1  46.487 

1  56.316 

2  06.146 

2  15.976 

2  25.805 

2  35.635 

50 

.137 

51 

1  26.992 

36.821 

1  46.651 

1  56.480 

2  06.310 

2  16.139 

2  25.969 

2  35.798 

51 

.139 

52 

1  27.155 

36.985 

1  46.815 

1  56.644 

2  06.474 

2  16.303 

2  26.133 

2  35.962 

52 

.142 

53 

1  27.319 

37.149 

46.978 

1  56.808 

2  06.637 

2  16.467 

2  26.297 

2  36.126 

53 

.145  H 

54 

1  27.483 

37.313 

47.142 

1  56.972 

2  06.801 

2  16.631 

2  26.460 

2  36.290  j 

54 

.147  I! 

55 

1  27.647 

1  37.476 

47.306 

1  57.136 

2  06.965 

2  16.795 

2  26.624 

2  36.454 

55 

.150 

56 

1  27.811 

1  37.640 

47.470 

1  57.299 

2  07.129 

2  16.959 

2  26.788 

2  36,618 

56 

.153  i 

57 

1  27.975 

1  37.804 

47.634 

1  57.463 

2  07.293 

2  17.122 

2  26.952 

2  36.781 

57 

.156 

58 

1  28.138 

I  37.968 

47.797 

1  57.627 

2  07.457 

2  17.286 

2  27.116 

2  36.945 

58 

.158 

59 

1  28.302 

1  38.132 

47.961 

1  57.791 

2  07.620 

2  17.450 

2  27.280 

2  37.109 

59 

.161 

85 


TABLE  IV.   SIDEREAL  INTO  MEAN  SOLAR  TIME. 


Side- 
real. 

16h- 

17h 

IS* 

19  h- 

20  h- 

21  b 

22  h- 

23  h 

For 

Seconds. 

ID. 

in.  ». 

m.  s 

m.  s. 

m.  s. 

m.  s. 

m.  s. 

m.  8. 

m.  r 

0 

2  37.273 

2*47.102 

2  56.932 

3  06.762 

3  16.591 

3  26.421 

3  36.250 

3  46.080 

B. 

1 

2  37.437 

2  47.266 

2  57.096 

3  06.925 

3  16.755 

3  26.585 

3  36.414 

3  46.244 

i 

0.003 

2 

2  37.601 

2  47.430 

2  57.260 

3  07.089 

3  16.919 

3  26.748 

3  36.578 

3  46.407 

2 

.005 

3 

2  37.764 

2  47.594 

2  57.424 

3  07.253 

3  17.083 

3  26.912 

3  36.742 

3  46.571 

3 

.008 

4 

2  37.928 

2  47.758 

2  57.587 

3  07.417 

3  17.246 

3  27.076 

3  36.906 

3  46.735 

4 

.011 

5 

2  38.092 

2  47.922 

2  57.751 

3  07.581 

3  17.410 

3  27.240 

3  37.069 

3  46.899 

5 

.014 

6 

2  38.256 

2  48.085 

2  57.915 

3  07.745 

3  17.574 

3  27.404 

3  37.233 

3  47.063 

6 

.016 

7 

2  38.420 

2  48.249 

2  58.079 

3  07.908 

3  17.738 

3  27.568 

3  37.397 

3  47.227 

7 

.019 

8 

2  38.584 

2  48.413 

2  58.243 

3  08.072 

3  17.902 

3  27.731 

3  37.561 

3  47,390 

8 

.022 

9 

2  38.747 

2  48.577 

2  58.406 

3  08.236 

3  18.066 

3  27.895 

3  37.725 

3  47.554 

9 

.025 

10 

2  38.911 

2  48.741 

2  58.570 

3  08.400 

3  18.229 

3  28.059 

3  37.889 

3  47.718 

10 

.027 

11 

2  39.075 

2  48.905 

2  58.734 

3  08.564 

3  18.393 

3  28.223 

3  38.052 

3  47.882 

11 

.030 

12 

2  39.239 

2  49.068 

2  58.898 

3  08.728 

3  18.557 

3  28.387 

3  38.216 

3  48.046 

12 

.033 

13 

2  39.403 

2  49.232 

2  59.062 

3  08.891 

3  18.721 

3  28.550 

3  38.380 

3  48.210 

13 

.035 

14 

2  39.566 

2  49.396 

2  59.226 

3  09.055 

3  18.885 

3  28.714 

3  38.544 

3  48.373 

14 

.038 

15 

2  39.730 

2  49.560 

2  59.389 

3  09.219 

3  19.049 

3  28.878 

3  38.708 

3  48.537 

15 

.041 

16 

2  39.894 

2  49.724 

2  59.553 

3  09.383 

3  19.212 

3  29.042 

3  38.871 

3  48.701 

16 

.044 

17 

2  40.058 

2  49.888 

2  59.717 

3  09.547 

3  19.376 

3  29.206 

3  39.035 

3  48.865 

17 

.046 

18 

2  40.222 

2  50.051 

2  59.881 

3  09.710 

3  19.540 

3  29.370 

3  39.199 

3  49.029 

18 

.049 

19 

2  40.386 

2  50.215 

3  00.045 

3  09.874 

3  19.704 

3  29.533 

3  39.363 

3  49.193 

19 

.052 

20 

2  40.549 

2  50.379 

3  00.209 

3  10.038 

3  19.868 

3  29:697 

3  39.527 

3  49.356 

20 

.055 

21 

2  40.713 

2  50.543 

3  00.372 

3  10.202 

3  20.032 

3  29.861 

3  39.691 

3  49.520 

21 

.057 

22 

2  40.877 

2  50.707 

3  00.536 

3  10.366 

3  20.195 

3  30.025 

3  39.854 

3  49.68-1 

22 

.060 

23 

2  41.041 

2  50.870 

3  00.700 

3  10.530 

3  20.359 

3  30.189 

3  40.018 

3  49.848 

23 

.063 

24 

2  41.205 

2  51.034 

3  00.864 

3  10.693 

3  20.523 

3  30.353 

3  40.182 

3  50.012 

24 

.066 

25 

2  41.369 

2  51.198 

3  01.028 

3  10.857 

3  20.687 

3  30.516 

3  40.346 

3  50.175 

25 

.068 

26 

2  41.532 

2  51.362 

3  01.192 

3  11.021 

3  20.851 

3  30.680 

3  40.510 

3  50.339 

26 

.071 

27 

2  41.696 

2  51.526 

3  01.355 

3  11.185 

3  21.014 

3  30.844 

3  40.674 

3  50.503 

27 

.074 

28 

2  41.860 

2  51.690 

3  01.519 

3  11.349 

3  21.178 

3  31.008 

3  40.837 

3  50.667 

28 

.076 

29 

2  42.024 

2  51.853 

3  01.683 

3  11.513 

3  21.342 

3  31.172 

3  41.001 

3  50.831 

29 

.079 

30 

2  42.188 

2  52.017 

3  01.847 

3  11.676 

3  21.506 

3  31.336 

3  41.165 

3  50.995 

30 

.082 

31 

2  42.352 

2  52.181 

3  02.011 

3  11.840 

3  21.670 

3  31.499 

3  41.329 

3  51.158 

31 

.085 

32 

2  42.515 

2  52.345 

3  02.174 

3  12.004 

3  21.834 

3  31.663 

3  41.493 

3  51.322 

32 

.087 

33 

2  42.679 

2  52.509 

3  02,338 

3  12.168 

3  21.997 

3  31.827 

3  41.657 

3  51.486 

33 

.090 

34 

2  42.843 

2  52.673 

3  *2.502 

3  12.332 

3  22.161 

3  31.991 

3  41.820 

3  51.650 

34 

.093 

!  35 

2  43.007 

2  52.836 

3  02.666 

3  12.496 

3  22.325 

3  32  155 

3  41.984 

3  51.814 

35 

.096 

36 

2  43.171 

2  53.000 

3  02.830 

3  12.659 

3  22.489 

3  32.318 

3  42.148 

3  51.978 

36 

.098 

37 

2  43.334 

2  53.164 

3  02.994 

3  12.823 

3  22.653 

3  32.482 

3  42.312 

3  52.141 

37 

.101 

38 

2  43i498 

2  53.328 

3  03.157 

3  12.987 

3  22.817 

3  32.646 

3  42.476 

3  52.305 

38 

.104 

39 

2  43.662 

2  53.492 

3  03.321 

3  13.151 

3  22.980 

3  32.810 

3  42.639 

3  52.469 

39 

.106 

40 

2  43.826 

2  53.656 

3  03v485 

3  13.315 

3  23.144 

3  32.974 

3  42.803 

3  52.633 

40 

.109 

41 

2  43.990 

2  53.819 

3  03,649 

3  13.478 

3  23.308 

3  33.138 

3  42.967 

3  52.797 

41 

.112 

42 

2  44.154 

2  53.983 

3  03.813 

3  13.642 

3  23.472 

3  33.301 

3  43.131 

3  52.961 

42 

.115 

43 

2  44.317 

2  54.147 

3  03.977 

3  13.»06 

3  23.636 

3  33.465 

3  43.295 

3  53.124 

43 

.117 

44 

2  44.481 

2  54.311 

3  04.140 

3  13.970 

3  23.800 

3  33.629 

3  43.459 

3  53.288 

44 

.120 

45 

2  44.645 

2  54.475 

3  04.304 

3  14.134 

3  23.963 

3  33.793 

3  43.622 

3  53.452 

45 

.123 

46 

2  44.809 

2  54.638 

3  04.468 

3  14.298 

3  24.127 

3  33.9:37 

3  43.786 

3  53.616 

46 

.126 

47 

2  44.973 

2  54.802 

3  04.632 

3  14.461 

3  24.291 

3  34.121 

3  43.950 

3  53.780 

47 

.128 

48 

2  45.137 

2  54.966 

3  04.796 

3  14.625 

3  24.455 

3  34.284 

3  44.114 

3  53.943 

48 

.131 

49 

2  45.300 

2  55.130 

3  04.960 

3  14.789 

3  24.619 

3  34.448 

3  44.278 

3  54.107 

49 

.134 

50 

2  45.464 

2  55.294 

3  05.123 

3  14.953 

3  24.782 

3  34.612 

3  44.442 

3  54.271 

50 

.137 

51 

2  45.628 

2  55.458 

3  05.287 

3  15.117 

3  24.946 

3  34.776 

3  44.605 

3  54.435 

51 

.139 

52 

2  45.792 

2  55.621 

3  05.451 

3  15.281 

3  25.110 

3  34.940 

3  44.769 

3  54.599 

52 

.142 

53 

2  45.956 

2  55.785 

3  05.615 

3  15.444 

3  25.274 

3  35.104 

3  44.933 

3  54.763 

53 

.145 

54 

2  46.120 

2  55.949 

3  05.779 

3  15.608 

3  25.438 

3  35.267 

3  45.097 

3  54.926 

54 

.147 

55 

2  46.283 

2  56.113 

3  05.942 

3  15.772 

3  25.602 

3  35.431 

3  45.261 

3  55.090 

55 

.150 

56 

2  46.447 

2  56.277 

3  06.106 

3  15.936 

3  25.765 

3  35.595 

3  45.425 

3  55.254 

56 

.153 

57 

2  46.611 

2  56.441 

3  06.270 

3  16.100 

3  25.929 

3  35.759 

3  45.588 

3  55.418 

57 

.156 

58 

2  46,775 

2  56.604 

3  06.434 

3  16.264 

3  26.093 

3  35.923 

3  45.752 

3  55.582 

58 

.158 

59 

2  46.939J 

8  fifi,768i  3  06.598 

8  16.427 

3  26.257 

3  36.Q86 

3  45.916 

3  55.746 

59 

.161 

86    • 


D.  Van  Nostrantf  s  Publications. 


Barre  Duparcq'§  Military  Art  and  History.  Elements  of 
Military  Art  and  History:  comprising  the  History  and  Tactics  of  the 
separate  Arms ;  the  Combination  of  the  Arms ;  and  the  minor  operations 
of  War.  By  EDWARD  DB  LA  BARRE  DUPARCQ,  Chef  de  Bataillon  of 
Engineers  in  the  Army  of  France ;  and  Professor  of  the  Military  Art  In 
the  Imperial  School  of  St.  Cyr.  Translated  by  Brig.-Gen.  GEO.  W.  OUL- 
LUM,  U.  S.  A.,  Chief  of  the  Staff  of  Major-Gen.  H.  W.  HALLECK,  General- 
in-Chief  U.  S.  Army.  1  vol.  8vo,  cloth.  $5.00. 

1 1  read  the  original  a  few  years  since,  and  considered  it  the  very  best  work  I  had 
seen  upon  the  subject.  Gen.  Cullum's  ability,  and  familiarity  with  the  technical  lan- 
guage of  French  military  writers,  are  a  sufficient  guarantee  of  the  correctness  of  his 
translation."  H.  "W.  HALLECK,  Major-Gen.  U.  S.  A. 

Bend's  Military  L,aw.  A  Treatise  on  Military  Law  and  the  Prac- 
tice of  Courts-Martial.  By  Capt.  S.  Y.  BENET,  Ordnance  Department, 
U.  S.  A.,  late  Assistant  Professor  of  Ethics,  Law,  &c.,  Military  Academy, 
West  Point.  Fifth  edition,  revised.  1  vol.  8vo,  law  sheep.  $4.50. 

"This  book  is  manifestly  well  timed  just  at  this  particular  period,  and  it  is,  without 
doubt,  quite  as  happily  adapted  to  the  purpose  for  which  it  was  written.  It  is  arranged 
with  admirable  method,  and  written  with  such  perspicuity,  and  in  a  style  so  easy  and 
graceful,  as  to  engage  the  attention  of  every  reader  who  may  be  so  fortunate  as  to  open 
its  pages.  This  treatise  will  make  a  valuable  addition  to  the  library  of  the  lawyer  or  the 
civilian;  while  to  the  military  man  it  seems  to  be  indispensable."— Philadelphia  Even- 
ing Journal. 

Halleck's  International  Law.  International  Law;  or,  Rules 
Regulating  the  Intercourse  of  States  in  Peace  and  War.  By  Major-Gen. 
H.  W.  HALLECK,  Commanding  the  Army.  1  vol.  8vo,  law  sheep.  $6.00. 

"  The  work  will  be  found  to  be  of  great  use  to  army  and  navy  officers,  to  professional 
lawyers,  and  to  all  interested  in  the  topics  of  which  it  treats — topics  to  which  present 
events  give  a  greatly  enhanced  importance— such  as  'Declaration  of  War  and  its 
Effects;1  'Sieges  and  Blockades;'  'Visitation  and  Search;1  'Right  of  Search;1  'Prize 
Courts;1  '  Military  Occupation  j1  'Treaties  of  Peace;1  ' Sovereignty  of  States,1  &c.,  and 
valuable  information  for  consuls  and  ambassadors.11—^.  T.  Evening  Post. 

History  of  West  Point.  And  its  Military  Importance  during  the 
American  Revolution ;  and  the  Origin  and  Progress  of  the  United  States 
Military  Academy.  By  Capt.  EDWARD  C.  BOYNTON,  A.  M.,  Adjutant  of 
the  Military  Academy.  With  numerous  Maps  and  Engravings.  1  v  ii 
8vo,  blue  cloth,  $6.00;  half  mor.,  $7.50;  full  mor.,  $10.00. 

"It  records  the  earliest  attempt  at  instituting  a  Military  School  by  the  Continental 
Congress,  in  1776.  It  conducts  us  through  the  life  of  the  institution,  arguing  with  terse- 
ness its  constitutionality,  defending  its  educational  principle8,»and  explaining  the  neces- 
sity for  its  preservation." — United  Service  Magazine. 

1 


D.  Van  Nostran&s  Publications. 

History  of  the  United  States  Naval  Academy.  With 
Biographical  Sketches,  and  the  Names  of  all  the  Superintendents,  Profes- 
sors, and  Graduates ;  to  which  is  added  a  Record  of  some  of  the  earliest 
votes  by  Congress,  of  Thanks,  Medals,  and  Swords  to  Naval  Officers. 
By  EDWARD  CHAUNCEY  MARSHALL,  A.  M.  1  vol.  12mo,  cloth,  plates.  $1.00. 

"Every  naval  man  will  find  it  not  only  a  pleasant  companion,  but  an  invaluable  book 
of  reference.  It  is  seldom  that  so  much,  information  is  made  accessible  in  so  agreeable  a 
manner  in  so  small  a  space."— New  York  Times. 

Scott's  Military  Dictionary.  Comprising  Technical  Definitions ; 
Information  on  Raising  and  Keeping  Troops ;  Actual  Service,  including 
Makeshifts  and  improved  Materiel,  and  Law,  Government,  Regulation, 
and  Administration  relating  to  Land  Forces.  By  Colonel  IL  L.  Scott,  In- 
spector-General U.  S.  A.  1  vol.  large  octavo,  fully  illustrated,  half  mo- 
rocco, $6;  halfrussia,  $8;  full  morocco,  $10. 

"  This  book  is  really  an  Encylopaedia,  both  elementary  and  technical,  and  as  such  occu- 
pies a  gap  in  military  literature  which  has  long  been  most  inconveniently  vacant.  This 
book  meets  a  present  popular  want,  and  will  be  secured  not  only  by  those  embarking  in 
the  profession  but  by  a  great  number  of  civilians,  who  are  determined  to  follow  the  do 
scriptions  and  to  understand  the  philosophy  of  the  various  movements  of  the  Campaign. 
Indeed,  no  tolerably  good  library  would  be  complete  without  the  work."— New  York 
Times. 

Ben  ton's  Ordnance  and  Gunnery.  A  Course  of  Instruction 
in  Ordnance  and  Gunnery;  compiled  for  the  use  of  the  Cadets  of  the 
United  States  Military  Academy.  By  Capt.  J.  G.  Benton,  Ordnance  De- 
partment, late  Instructor  of  Ordnance  and  Gunnery,  Military  Academy, 
"West  Point ;  Principal  Assistant  to  Chief  of  Ordnance,  U.  S.  A.  Third 
Edition,  revised  and  enlarged.  1  vol.  8vo,  cloth,  cuts.  $5.00. 

"There  is  no  one  book  within  the  range  of  our  military  reading  and  study,  that  con- 
tains more  to  recomtnend  it  upon  the  subject  of  which  it  treats.  It  is  as  full  and  com- 
plete as  the  narrow  compass  of  a  single  volume  would  admit,  and  the  reputation  of  the 
author  as  a  scientific  and  practical  artillerist  is  a  sufficient  guarantee  for  the  correctness 
of  his  statements  and  deductions,  and  the  thoroughness  of  his  labors." — N.  Y.  Observer. 

Gibbon's  Artillerist's  Ulanual.  Compiled  from  various  Sources, 
and  adapted  to  the  Service  of  the  United  States.  Profusely  illustrated 
with  woodcuts  and  engravings  on  stone.  Second  edition,  revised  and 
corrected,  with  valuable  additions.  By  Gen.  John  Gibbon,  U.  S.  A. 
1  vol.  8vo,  half  roan.  $6.00. 

This  book  is  now  considered  the  standard  authority  for  that  particular  branch  of  the 
Service  in  the  United  States  Army.  The  War  Department,  at  Washington,  has  exhibited 
its  thorough  appreciation  of  the  merits  of  this  volume,  the  want  of  which  has  been  hither- 
to much  felt  in  the  service,  by  subscribing  for  700  copies. 

2 


D.  Van  No str and' s  Publications. 

Jo  ma  ail's  Life  of  tlic  Emperor  Xapolcoa  I.  Life  of  Napo- 
leon. By  Baron  Joraini,  General-in-Chief  and  Aide-de-Camp  to  the 
Emperor  of  Russia.  Translated  from  the  French,  with  Notes,  by  H.  "W. 
Halleck,  LL.  D.,  Major-General  United  States  Army;  author  of  "Elements 
of  Military  Art  and  Science,"  "  International  Law,  and  the  Laws  of  War,'' 
etc.,  etc.  In  four  volumes  octavo,  with  an  Atlas  of  Sixty  Maps  and 
Plans.  Price,  in  red  cloth,  $25.00;  half  calf,  or  half  morooco,  $35.00; 
half  russia,  $37.50. 

u  It  is  needless  to  say  any  tiling  in  praise  of  Jomini  as  a  writer  on  the  science  of  war. 

"  General  Halleck  has  laid  the  professional  soldier  and  the  student  of  military  history 

under  equal  obligations  by  the  service  he  has  done  to  the  cause  of  military  literature  in 

the  preparation  of  this  work  for  the  press.    His  rare  qualifications  for  the  task  thus 

undertaken  will  be  acknowledged  by  all. 

"The  Notes  with  which  the  text  is  illustrated  by  General  Halleck  are  not  among  the 
least  of  the  merits  of  the  publication,  which,  in  this  respect,  has  a  value  not  possessed  by 
the  original  work.  *  *  *"  — National  Intelligencer. 

"  The  Atlas  attached  to  this  version  of  Jominfs  Napoleon  adds  very  materially  to  its 
value.  It  contains  sixty  Maps,  illustrative  of  Napoleon's  extraordinary  military  career, 
beginning  with  the  immortal  Italian  Campaigns  of  1796,  and  closing  with  the  decisive 
Campaign  of  Flanders,  in  1815,  the  last  Map  showing  the  Battle  of  Wavre.  These  Maps 
take  the  reader  to  Italy,  Egypt,  Palestine,  Germany  Moravia,  Russia,  Spain,  Portugal, 
and  Flanders;  and  their  number  and  variety,  and  the  vast  and  various  theatres  of  action 
which  they  indicate,  testify  to  the  immense  extent  of  Napoleon's  operations,  and  to  the 
gigantic  character  of  his  power.  *  *  *"  —Boston  Traveller. 

Jomiiii's  Grand  Military  Operations.  Treatise  on  Grand 
Military  Operations ;  or,  a  Critical  and  Military  History  of  the  "Wars  of 
Frederick  the  Great,  as  contrasted  with  the  Modern  System,  together 
with  a  few  of  the  most  important  Principles  of  the  Art  of  War.  By 
Baron  Jomini,  General-in-Chief  and  Aide-de-Camp  to  the  Emperor  of 
Russia.  Translated  from  the  French  by  Col.  S.  B.  Holabird,  A.  D.  C. 
U.  S.  A.  2  vols.  8vo,  cloth,  with  an  Atlas  of  40  Maps  and  Plans.  $15.00. 

Joniini's  Campaign  of  Waterloo.     The  Political  and  Military 
History  of  the  Campaign  of  Waterloo.     Translated  from  the  French  of 
General  Baron  de  Jomini,  by  Capt.  S.  V,  Benet,  Ordnance  Department, 
U.  S.  Army.     Third  edition.     1  vol.  12mo,  cloth,  $1.25. 
"Baron  Jomini  has  the  reputation  of  being  one  of  the  greatest  military  historians  and 
critics  of  the  century.     His  merits  have  been  recognized  by  the  highest  military  author- 
ities in  Europe,  and  were  rewarded  in  a  conspicuous  manner  by  the  greatest  military 
power  in  Christendom.    He  learned  the  art  of  war  in  the  school  of  experience,  the  best 
and  only  finishing  school  of  the  soldier.    He  served  with  distinction  in  nearly  all  the 
campaigns  of  Napoleon,  and  it  was  mainly  from  the  gigantic  military  operations  of  this 
matchless  master  of  the  art  that  he  was  enabled  to  discover  its  true  principles,  and  to 
ascertain  the  best  means  of  their  application  to  the  infinity  of  combinations  which  actual 
war  presents.    Jomini  criticises  the  details  of  "Waterloo  with  great  science,  and  yet  in  a 
manner  that  interests  the  general  reader  as  well  as  the  professional."— New  York  World. 

3 


D.  Van  Nostrand's  Publications. 


Rocmer's  Cavalry;  its  History,  Management,  and 
Uses  ill  War.  By  J.  Roemer,  LL.  D.,  late  an  Officer  of  Cavalry  in 
the  Service  of  the  Netherlands.  Elegantly  illustrated,  with  one  hundred 
and  twenty-seven  fine  wood  engravings.  In  one  large  octavo  volume, 
beautifully  printed  on  tinted  paper.  Cloth,  $6;  half  calf,  $7.50. 

"  By  far  the  best  treatise  upon  Cavalry  and  its  uses  in  the  field,  which  has  yet  been 
published  in  this  country,  for  the  general  use  of  officers  of  all  ranks,  is  this  elaborate  and 
interesting  work.  Eschewing  the  elementary  principles  and  tactics  of  cavalry,  which 
may  be  learned  from  any  hand-book,  the  author  treats  of  the  uses  of  cavalry  in  the  field 
of  strategy  and  tactics,  and  of  its  general  discipline  and  management.  The  range  of  the 
work  includes  an  admirable  treatise  upon  rifled  fire-arms,  an  historical  sketch  of  cavalry, 
embodying  many  interesting  facts,  an  account  of  the  cavalry  service  in  Europe  and  this 
country,  and  a  treatise  on  horses,  their  equipment,  management,  &c.  The  work  is  copi- 
ously illustrated  and  elegantly  printed.  It  is  interesting  not  alone  to  military  men  but 
to  the  general  reader,  who  will  gain  from  its  pages  valuable  historical  facts  and  very  clear 
ideas  of  some  branches  of  the  art  of  war,  such  as  the  employment  of  spies,  gaining  infor- 
mation in  an  enemy's  country,  advance  movements,  and  other  strategical  manoeuvres." — 
Boston  Journal. 

Nolan's  System  for  Training  Cavalry  Horses.  By  Ken- 
ner  Garrard,  Captain  Fifth  Cavalry,  U.  S.  A.  1  vol.  12mo,  cloth.  24 
lithographed  plates.  $2.00. 

"  This  work  is  clearly  written,  is  eminently  practical,  is  fully  illustrated,  and  contains 
numerous  hints  as  applicable  to  the  discipline  and  management  of  the  draught-horse  us 
that  ofiis  more  showy  and  fiery  brother  of  the  cavalry."— Boston  Journal. 

Barnard  and  Barry's  Peninsular  Campaign.  Report  of 
the  Engineer  and  Artillery  Operations  of  the  Army  of  the  Potomac,  from 
its  organization  to  the  close  of  the  Peninsular  Campaign.  By  Brig.-Gen. 
J.  G.  Barnard,  and  other  Engineer  Officers,  and  Brig.-Gen.  W.  F.  Barry, 
Chief  of  Artillery.  Illustrated  by  numerous  Maps,  Plans,  etc.  1  vol. 
8vo,  cloth,  $4.00. 

"  The  title  of  this  work  sufficiently  indicates  its  importance  and  value  as  a  contribu- 
tion to  the  history  of  the  great  rebellion.  General  Barnard's  Eeport  is  a  narrative  of  the 
Engineer  operations  of  the  Array  of  the  Potomac  from  the  time  of  its  organization  to 
the  date  it  was  withdrawn  from  the  James  Eiver.  Thus  a  record  is  given  of  an  impor- 
tant part  in  the  great  work  which  the  nation  found  befoie  it  when  it  was  first  confronted 
with  the  necessity  of  war;  and  perhaps  on  no  other  point  in  the  annals  of  the  rebellion 
will  future  generations  look  with  a  deeper  or  more  admiring  interest."— Buffalo  Courier. 

The  "  C.  S.  A.,"  and  the  Battle  of  Bull  Run.  (A  Let- 
ter to  an  English  friend),  by  J.  G.  Barnard,  Lieut.-Colonel  of  Engineers, 
U.  S.  A.,  Brigadier-General  and  Chief  Engineer,  Army  of  the  Potomac. 
"With  five  Maps.  1  vol.  8vo,  cloth.  $2.00. 

"The  work  is  clearly  written,  and  can  but  leave  the  impression  upon  every  readers 
mind  that  it  is  truth.  We  commend  it  to  the  perusal  of  every  one  who  wants  an  intelli- 
gent, truthful,  and  graphic  description  of  the  '  C.  S.  A.,'  and  the  Battle  of  Bull  Run."— 
New  York  Observer. 

4 


D.  Van  No  sir  and}  s  Publications. 

Simpson's  Ordnance  and  Naval  Gunnery.  A  Treatise  on 
Ordnance  and  Naval  Gunnery,  compiled  and  arranged  as  a  Text-Book  for 
the  U".  S.  Naval  Academy.  By  Lieut.  Edward  Simpson,  U.  S.  N.  Third 
edition,  revised  and  enlarged.  1  vol.  Svo,  plates  and  cuts,  cloth.  $5.00. 

*'  I';  3  scarcely  necessary  for  us  to  say,  that  a  work  prepared  by  a  writer  so  practically 
conversant  with  all  the  subjects  of  which  he  treats,  and  who  has  such  a  reputation  for 
scientific  ability,  cannot  fail  to  take  at  once  a  high  place  among  the  text-books  01  our 
naval  service.  It  has  been  approved  by  the  Secretary  of  the  Navy,  and  will  henceforth 
be  one  of  the  standard  authorities  on  all  matters  connected  with  Naval  Gunnery."— New 
York  Herald. 

Holley's  Ordnance  and  Armor.  Embracing  Descriptions, 
Discussions,  and  Professional  Opinions  concerning  the  Material,  Fabrica- 
tion, Requirements,  Capabilities,  and  Endurance  of  European  and  Amer- 
ican Guns  for  Naval,  Sea-Coast,  and  Iron-Clad  Warfare,  and  their  Rifling, 
Projectiles,  and  Breech-Loading;  also,  Results  of  Experiments  against 
Armor,  from  Official  Records.  With  an  Appendix,  referring  to  Gun- 
Cotton,  Hooped  Guns,  etc.,  etc.  By  A.  L.  Holley,  B.  P.  With  4.93  illus- 
trations. 1  vol.  8vo,  948  pages.  Half  roan,  $10.00. 

Luce's  Naval  Light  Artillery.  Instructions  for  Naval  Light 
Artillery,  afloat  and  ashore,  prepared  and  arranged  for  the  U.  S.  Naval 
Academy,  by  Lieutenant  W.  H.  Parker,  U.  S.  N.  Second  edition,  revised 
by  Lieutenant  S.  B.  Luce,  U.  S.  N.,  Assistant  Instructor  of  Gunnery  and 
Tactics  at  the  U.  S.  Naval  Academy.  1  vol.  8vo,  cloth,  with  twenty-two 
Plates.  $3.00. 

"The  service  for  which  this  is  the  text-book  of  instruction  is  of  special  importance  in 
the  present  war.  The  use  of  light  boat-pieces  is  constant  and  important,  and  young  offi- 
cers are  frequently  obliged  to  leave  their  boats,  take  their  pieces  ashore,  and  manoeuvre 
them  as  field  artillery.  Not  unfrequently,  also,  they  are  incorporated,  when  ashore,  with 
troops,  and  must  handle  their  guns  like  the  artillery  soldiers  of  a  battery.  '  The  Exercise 
of  the  Howitzer  Afloat1  was  prepared  and  arranged  by  Captain  Dahlgren,  whose  name 
gives  additional  sanction  and  value  to  the  book.  A  Manual  for  the  Sword  and  Pistol  is 
also  given.  The  Plates  are  numerous  and  exceedingly  clear,  and  the  whole  typography  ijj 
excellent."— Philadelphia  Inquirer. 

Ward's  Waval  Ordnance  and  Gunnery.    Elementary  In- 
struction in  Naval  Ordnance  and  Gunnery.    By  James  H.  Ward,  Com- 
mander U.  S.  Navy ;  author  of  "  Naval  Tactics,"  and  "  Steam  for  the 
Million."    New  edition,  revised  and  enlarged.     8vo,  cloth.    $2.00. 
u  It  conveys  an  amount  of  information  in  the  same  space  to  be  found  nowhere  else,  and 

given  with  a  clearness  which  renders  it  useful  as  well  to  the  general  as  the  professional 

inquirer."— N.  Y.  Evening  Post. 
"  The  whole  detail  of  Ordnance,  In    ts  history,  philosophy,  and  application,  is  glvea 

by  Commander  "Ward  in  such  a  manner  (with  occasional  diagrams)  as  to  convey  to  the 

student  accurate  notions  for  practical  use."— New  Yorker. 

5 


D.  Van  No  sir  and}  s  Publications. 


I 

lance's   Seamanship.     Compiled  from  various  authorities,  and  Illus- 
trated with  numerous  original  and  selected  Designs.     For  the  use  of  the 
United  States  Naval  Academy.    By  S.  B.  Luce,  Lieut.-Commander  U.  S.  N. 
j          In  two  parts.    Second  Edition.  ,  One  royal  octavo  volume,  cloth.    $10.00. 

Squadron  Tactics  Under  Steam.  By  Foxhall  A.  Parker, 
Commander  U.  S.  Navy.  Published  by  authority  of  the  Navy  Depart- 
ment. 1  voL  8vo,  with  numerous  Plates.  $5.00. 

"In  this  useful  work  to  Navy  officers,  the  author  demonstrates — by  the  aid  of  profuse 
diagrams  and  explanatory  text — a  new  principle  for  manoeuvring  naval  vessels  in  action. 
The  author  contends  that  the  winds,  waves,  and  currents  of  the  ocean  oppose  no  more 
serious  obstacles  to  the  movements  of  a  steam  fleet,  than  do  the  inequalities  on  the  sur- 
face of  the  earth  to  the  manoeuvres  of  an  army.  It  is  in  this  light,  therefore,  that  he 
views  a  vast  fleet— simply  as  an  army ;  the  regiments,  brigades,  and  divisions  of  which 
are  represented  by  a  certain  ship  or  ships." — Scientific  American. 

Nautical  Routine  and  Stowage.  With  Short  Rules  in  Navi- 
gation. By  John  McLeod  Murphy  and  "Wm.  N.  Jeffers,  Jr.,  U.  S.  Navy. 
1  vol.  8vo,  blue  cloth.  $2.50. 

O§bon'§  Hand-Book  of  the  United  States  Navy.  Be- 
ing a  Compilation  of  all  of  the  Principal  Events  in  the  History  of  every 
Vessel  of  the  United  States  Navy,  from  April,  1861,  to  May,  1864. 
Compiled  and  arranged  by  B.  S.  Osbon.  1  vol.  12mo,  blue  cloth.  $2.50. 

"  Ae  a  condensed  and  compact  history,  as  well  as  a  work  containing  a  vast  amount  of 
Information,  this  work  cannot  be  surpassed." — Boston  Traveller. 

Brandt's  Gunnery  Catechism.  Gunnery  Catechism,  as  applied 
to  the  Service  of  Naval  Ordnance.  Adapted  to  the  Latest  Official  Regu- 
lations, and  approved  by  the  Bureau  of  Ordnance,  Navy  Department. 
By  J.  D.  Brandt,  formerly  of  the  U.  S.  Navy.  1  vol.  18mo,  illustrated, 
blue  cloth.  $1.50. 

"This  manual  is  very  full  of  information  and  instruction,  and  shows  the  'chief  end'  of 
Gunnery,  and  the  aim  of  those  who  follow  that  profession.  It  is  indispensable  to  all 
those  who  are  suddenly  introduced  to  a  gun-deck,  and  will  be  found  a  valuable  aid  also 
to  experienced  officers."—  Commercial  Advertiser . 

Barrett's  Gunnery  Instructions.  Gunnory  Instruction^,  sim- 
plified for  the  Volunteer  Officers  of  the  United  States  Navy,  with  Hints 
to  Executive  and  other  Officers.  By  Lieut.  Edward  Barrett,  U.  S.  N., 
Instructor  of  Gunnery,  Navy  Yard,  Brooklyn.  1  vol.  12mo,  cloth.  $1.25. 

"It  is  a  thorough  work,  treating  plainly  on  its  subject  and  contains  also  somfi  valu- 
able hints  to  executive  officers.  No  officer  In  the  volunteer  navy  should  be  without  a 
coj  y  "—Boston  Evening  Traveller. 


D.  Van  No  sir  and?  s  Publications. 

Totten's  Naval  Text-Book.  Naval  Text-Book  and  Dictionary, 
compiled  for  the  use  of  the  Midshipmen  of  the  U.  S.  Navy.  By  Com 
mander  B.  J.  Totten,  U.  S.  N.  Second  and  revised  edition.  1  vol.  12mo. 
$3.00. 

Calculated  Tables  of  Ranges  for  Navy  and  Army 
Guns.  With  a  Method  of  Finding  the  Distance  of  an  Object  at  Sea. 
By  Lieut.  W.  P.  Buckner,  U.  S  N.  1  vol.  8vo,  cloth.  $1.50. 


JUanual  of  Internal  Rules  and  Regulations  for  Men- 
of- War.  By  Commodore  U.  P.  Levy,  U.  S.  N.,  late  Flag  Officer  com- 
manding U.  S.  Naval  Force  in  the  Mediterranean,  &c.  Flexible  blue 
cloth.  Third  Edition,  revised  and  enlarged.  50  cents. 

"Among  the  professional  publications  for  which  we  are  indebted  to  the  war,  we  will- 
ingly give  a  prominent  place  to  this  useful  little  Manual  of  Eules  and  Regulations  to  be 
observed  on  board  of  ships  of  war.  Its  authorship  is  a  sufficient  guarantee  for  its  accu- 
racy and  practical  value ;  and  as  a  guide  to  young  officers  in  providing  for  the  discipline, 
police,  and  sanitary  government  of  the  vessels  under  their  command,  we  know  of  nothing 
superior."— N.  Y.  Herald. 


King's   Lessons   and   Practical  Notes   on   Steam,  The 

Steam  Engine,  Propellers,  &c.,  &c.,  for  Young  Marine  Engineers,  Stu- 
dents, and  others.  By  the  late  W.  H.  King,  U.  S.  Navy.  Revised  by 
Chief  Engineer  J.  W.  King,  U.  S.  Na\/.  Ninth  Edition,  enlarged.  8vo, 
cloth.  $2.00. 

"  This  is  the  ninth  edition  of  a  valuable  work  of  the  late  W.  HTKing,  U.  S.  Navy.  It 
contains  lessons  and  practical  notes  on  Steam  and  the  Steam-Engine,  Propellers,  &c.  It 
is  calculated  to  be  of  great  use  to  young  marine  engineers,  students,  and  others.  The 
text  is  illustrated  and  explained  by  numerous  diagrams  and  representations  of  machi- 
nery. This  new  edition  has  been  revised  and  enlarged  by  Chief  Engineer  J.  W.  King, 
U.  8.  Navy,  brother  to  the  deceased  author  of  the  work."— Boston  Daily  Advertiser. 


Ward's  Steam  for  the  Million.  A  popular  Treatise  on  Steam 
and  its  Application  to  the  Useful  Arts,  especially  to  Navigation.  By 
J.  H.  "Ward,  Commander  U.  S.  Navy.  New  and  revised  Edition.  1  vol. 
8vo,  cloth.  $1.00. 

The  Naval  Howitzer  Ashore.  By  Foxhall  A.  Parker,  Com- 
mander U.  S.  Navy.  1  vol.  8vo,  with  Plates.  Cloth.  $4.00. 


D.  Van  No  sir  and}  s  Publications. 


Gillmore's  Fort  S muter.  Official  Report  of  Operations  against 
the  Defences  of  Charleston  Harbor,  18G3.  Comprising  the  Descent  upon 
Morris  Island,  the  Demolition  of  Fort  Sumter,  and  the  Siege  and  Reduc- 
tion of  Forts  Wagner  and  Gregg.  By  Major-Gen.  Q.  A.  GILLMORE,  U.  S. 
Volunteers,  and  Major  U.  S.  Corps  of  Engineers.  With  Maps  and  Litho- 
graphic Plates,  Views,  &c.  1  vol.  8vo.  cloth.  $10. 

Gillmore's  Siege  and  Reduction  Of  Fort  Pulaski, 
Georgia.  Papers  on  Practical  Engineering.  No.  8.  Official  Report 
to  the  U.  S.  Engineer  Department  of  the  Siege  and  Reduction  of  Fort 
Pulaski,  Ga.,  February,  March,  and  April,  1862.  By  Brig.-Gen.  Q.  A. 
GILLMORE,  U.  S.  A.  Illustrated  by  Maps  and  Views.  1  vol.  8vo,  cloth. 
$2.50. 

"This  Is  an  official  history  of  the  siege  of  Fort  Pulaski,  from  the  commencement,  with 
all  the  details  in  full,  made  up  from  a  daily  record,  forming  a  most  valuable  paper  for 
future  reference.  The  situation  and  construction  of  the  Fort,  the  position  of  the  guns, 
both  of  the  rebels  and  the  Federals,  and  their  operation,  are  made  plain  by  maps  and 
engraved  views  of  different  sections.  Additional  reports  from  other  officers  are  fur- 
nished in  the  appendix,  and  every  thing  has  been  done  to  render  the  work  full  and  reli- 
able."— Boston  Journal. 

Gill  in  ore's  Treatise  on  Limes,  Hydraulic  Cements,  and 
Mortars.  Papers  on  Practical  Engineering,  U.  S.  Engineer  Depart- 
ment, No.  9,  containing  Reports  of  numerous  Experiments  conducted  in 
New  York  City,  during  the  years  1858  to  1861,  inclusive.  By  Q.  A. 
GILLMORE,  Brig.-Gen.  U.  S.  Volunteers,  and  Major  U.  S.  Corps  of  Engi- 
neers. With  numerous  Illustrations.  One  vol.  8vo.  $4.00. 
"  This  work  contains  a  record  of  certain  experiments  and  researches  made  under  the 

authority  of  the  Engineer  Bureau  of  the  War  Department  from  1858  to  1861,  upon  the 

various  hydraulic  cements  of  the  United  States,  and  the  materials  for  their  manufacture. 

The  experiments  were  carefully  made,  and  are  well  reported  and  compiled.'' — Journal 

Franklin  Institute. 

Tbe  Volunteer  Quartermaster.  Containing  a  Collection  and 
Codification  of  the  Laws,  Regulations,  Rules,  and  Practices  governing  the 
Quartermaster's  Department  of  the  United  States  Army,  and  in  force 
March  4,  1865.  By  Captain  ROELIFP  BRINKERHOFF,  Assistant  Quarter- 
master U.  S.  Volunteers,  and  Post  Quartermaster  at  "Washington.  1 
vol.  12mo,  cloth.  $2.50. 

This  work  embraces  all  the  laws  of  Congress,  and  all  the  orders  and  circulars  of  the 
War  Office  and  its  bureaus,  bearing  upon  the  subject  It  also  embodies  the  decisions  of 
the  Second  Comptroller  of  the  Treasury,  so  far  as  they  affect  the  Quartermaster's  Depart- 
ment These  decisions  have  the  force  of  law  in  the  adjustment  of  accounts,  and  are  there- 
fore invaluable  to  all  disbursing  officers. 


D.  Van  No  sir  and}  s  Publications. 

1  ill hnii's  Military  Bridges.  Systems  of  Military  Bridges  in  Use 
by  the  United  States' Army;  those  adopted  by  the  Great  European 
Powers ;  and  such  as  are  employed  in  British  India.  "With  Directions 
for  the  Preservation,  Destruction,  and  Re-establishment  of  Bridges.  By 
Brig.-Gen.  GEORGE  "W.  CULLUM,  Lieut.-Col.  Corps  of  Engineers,  U.  S.  A. 
1  vol.  8vo.  "With  numerous  Illustrations,  cloth.  $3.50. 

44  We  have  no  man  more  competent  to  prepare  such  a  work  than  Brig.-Gen.  Cullum, 
who  had  the  almost  exclusive  supervision,  devising,  building,  and  preparing  for  service 
of  the  various  bridge-trains  sent  to  our  armies  in  Mexico  during  our  war  with  that 
country.  The  treatise  before  us  is  very  complete,  and  has  evidently  been  prepared  with 
scrupulous  care.  The  descriptions  of  the  various  systems  of  military  bridges  adopted 
by  nearly  all  civilized  nations  are  very  interesting  even  to  the  non-professional  reader, 
and  to  those  specially  interested  in  such  subjects  must  be  very  instructive,  for  they  are 
evidently  the  work  of  a  master  of  the  art  of  military  bridge-building." —  Washington 
Chronicle. 

llaupt's  Military  Bridges.  For  the  Passage  of  Infantry,  Artil- 
lery, and  Baggage-Trains ;  with  Suggestions  of  many  new  Expedients 
and  Constructions  for  Crossing  Streams  and  Chasms ;  designed  to  utilize 
the  Resources  ordinarily  at  command,  and  reduce  the  amount  and  cost  of 
Army  Transportation.  Including  also  Designs  for  Trestle  and  Truss 
Bridges  for  Military  Railroads,  adapted  especially  to  the  wants  of  the 
Service  of  the  United  States.  By  HERMAN  HAUPT,  Brig.-Gen.  in  charge 
of  the  Construction  and  Operation  of  the  U.  S.  Military  Railways,  Author 
of  "  General  Theory  of  Bridge  Construction,"  &c.  Illustrated  by  sixty- 
nine  Lithographic  Engravings.  8vo,  cloth.  $6.50. 

"This  elaborate  and  carefully  prepared,  thouth  thoroughly  practical  and  simple  work, 
is  peculiarly  adapted  to  the  military  service  of  the  United  States.  Mr.  Haupt  has  added 
very  much  to  the  ordinary  facilities  for  crossing  streams  and  chasms,  by  the  instructions 
afforded  in  this  work."— Boston  Courier. 

Holley's  Railway  Practice.  American  and  European  Railway 
Practice,  in  the  Economical  Generation  of  Steam,  including  the  Materials 
and  Construction  of  Coal-burning  Boilers,  Combustion,  the  Variable 
Blast,  Vaporization,  Circulation,  Superheating,  Supplying  and  Heating 
Feed- water,  &c.,  and  the  Adaptation  of  "Wood  and  Coke-burning  Engines 
to  Coal-burning ;  and  in  permanent  "Way,  including  Road-bed,  Sleepers, 
Rails,  Joint  Fastenings,  Street  Railways,  &c.,  &c.  By  Alexander  L. 
Holley,  B.  P.  "With  77  lithographed  plates.  1  vol.  folio,  cloth.  $12.00. 

*  *  *  "  All  these  subjects  are  treated  by  the  author  in  both  an  intelligent  and  in- 
telligible manner.  The  facts  and  ideas  are  well  arranged,  and  presented  in  a  clear  and 
simple  style,  accompanied  by  beautiful  engravings,  and  we  presume  the  work  will  be 
regarded  as  indispensable  by  all  who  are  interested  in  a  knowledge  of  the  construction 
of  railroads,  and  rolling  stock,  or  the  working  of  locomotives.11— Scientific  American. 

9 


D.  Van  Nostrand?  s  Publications. 


Authorized  U.  S.  Infantry  Tactics.  For  the  Instruction, 
Exercise,  and  Manoeuvres  of  the  Soldier,  a  Company,  Line  of  Skirmish, 
ers,  Battalion.  Brigade,  or  Corpa  d'Armee.  By  Brig.-Gen.  SILAS  CASEY, 
U  S.  A.  3  vols  24mo.  Cloth,  lithographed  plates.  $2.50. 

Vol.  I. — School  of  the  Soldier;  School  of  the  Company;  Instruction  for 
Skirmishers. 

Yol.  II.— School  of  the  Battalion. 

Yol.  III. — Evolutions  of  a  Brigade ;  Evolutions  of  a  Corps  d'Armee. 

"WAR  DEPARTMENT,  WASHINGTON,  August  11, 1S62. 

"  The  System  of  Infantry  Tactics  prepared  by  Brig.-Gen.  Silas  Casey,  U.  8.  A.,  having 
been  approved  by  the  President,  is  adopted  for  the  instruction  of  the  Infantry  of  the 
Armies  of  the  United  States,  whether  Regular,  Volunteer,  or  Militia. 

"EDWIN  M.  STANTON,  Secretary  of  Far." 

U.  S.  Tactics  for  Colored  Troops.  U.  S.  Infantry  Tactics, 
for  the  Instruction,  Exercise,  and  Manoeuvres  of  the  Soldier,  a  Company, 
Line  of  Skirmishers,  and  Battalion,  for  the  use  of  the  Colored  Troops  of 
the  United  States  Infantry.  Prepared  under  the  direction  of  the  War 
Department.  I  vol.,  plates.  $1.50. 

"WAR  DEPARTMENT,  WASHINGTON,  March  9, 1863. 

"  This  system  of  United  States  Infantry  Tactics,  prepared  under  the  direction  of  the 
War  Department,  for  the  use  of  the  Colored  Troops  of  the  United  States  Infantry,  having 
been  approved  by  the  President,  is  adopted  for  the  instruction  of  such  troops. 

"EDWIN  M.  STANTON,  Secretary  of  War." 

Keltoii's  Bayonet  Exercise.  A  New  Manual  of  the  Bayonet, 
for  the  Army  and  Militia  of  the  United  States.  By  Colonel  J.  C.  KELTON, 
U.  S.  A.  With  forty  beautifully  engraved  plates.  Red  cloth.  $2.00. 

"This  Manual  was  prepared  for  the  use  of  the  Corps  of  Cadets,  and  has  been  introduced 
at  the  Military  Academy  with  satisfactory  results.  It  is  simply  the  theory  of  the  attack 
and  defence  of  the  sword  applied  to  the  bayonet,  on  the  authority  of  men  bkilled  in  the 
use  of  arms." — New  York  Times. 

JBcrriniau's  Sword-Play.  The  Militiaman's  Manual  and  Sword- 
Play  without  a  Master. — Rapier  and  Broad-Sword  Exercises  copiously 
Explained  and  Illustrated ;  Small- Arm  Light  Infantry  Drill  of  the  United 
States  Army ;  Infantry  Manual  of  Percussion  Muskets ;  Company  Drill 
of  the  United  States  Cavalry.  By  Major  M.  \V.  BEftRiMAN,  engaged  for 
the  last  thirty  years  in  the  practical  instruction  of  Military  Students. 
Fourth  edition.  1  vol.  12mo,  red  cloth.  $1.00. 

"This  work  will  be  found  very  valuable  to  all  persons  seeking  military  instruction; 
but  it  recommends  itself  most  especially  to  officers,  and  those  who  have  to  .ise  the  sword 
or  sabre.  We  believe  it  is  the  only  work  on  the  use  of  the  sword  published  in  ,nis 
country."— New  York  Tablet. 

10 


D.  Van  No  sir  and}  s  Publications. 

i 

Heavy  Artillery  Tactics.— 1863.  Instruction  for  Heavy  Ar. 
tillery ;  prepared  by  a  Board  of  Officers,  for  the  use  of  the  Army  of  the 
United  States.  With  service  of  a  gun  mounted  on  an  iron  carriage.  In 
one  volume,  12 mo,  with  numerous  illustrations,  cloth.  $2.50. 

"WAR  DEPARTMENT,  > 

"WASHINGTON,  D.  C.,  Oct.  20,  1862.  ) 

"This  system  of  Heavy  Artillery  Tactics,  prepared  under  direction  of  the  War  De- 
partment, having  been  approved  by  the  President,  is  adopted  for  the  instruction  of 
troops  when  acting  as  heavy  artillery.  EDWIN  M.  STANTON,  Secretary  of  War," 

w  The  First  Part  consists  of  sixteen  lessons  relating  to  the  service  of  the  single  piece,  in- 
cluding the  gun,  howitzer,  mortar,  and  columbiad ;  also,  the  formation  of  batteries,  the  art 
of  aiming  pieces  and  firing  hot-shot.  Part  Second  relates  entirely  to  mechanical  manoeu- 
vres, and  appliances  for  handling,  mounting,  dismounting,  and  transporting  heavy  pieces. 
Part  Third  is  of  a  miscellaneous  cnaracter,  containing  directions  for  embarking  and  disem- 
barking artillery  and  ordnance  stores;  also,  tables  of  dimensions  and  weights  of  guns, 
carriages,  shot,  shell,  machines,  and  implements,  with  charges  for,  and  ranges  of  heavy 
artillery.  These  instructions  are  not  only  copious  in  detail,  but  aptly  illustrated  with 
thirty-nine  elegant  steel-plate  engravings." — Bulletin. 

Roberta's  Hand-Book  of  Artillery.  For  the  Service  of  the 
United  States  Army  and  Militia.  New  revised  and  greatly  enlarged 
edition.  By  Major  JOSEPH  ROBERTS,  U.  S.  A.  1  vol.  ISmo,  cloth. 

$1.25. 

"A  complete  catechism  of  gun  practice,  covering  the  whole  ground  of  this  branch  of 
military  science,  and  adapted  to  militia  and  volunteer  drill,  as  well  as  to  the  regular 
army.  It  has  the  merit  of  precise  detail,  even  to  the  technical  names  of  all  parts  of  a 
gun,  and  how  the  smallest  operations  connected  with  its  use  can  be  best  performed.  It 
has  evidently  been  prepared  with  great  care,  and  with  strict  scientific  accuracy." 
New  York  Century. 

Duauc's  Manual  for  Engineer  Troops.  Consisting  of  Part 
I.,  Pontoon  Drill ;  II.,  Practical  Operations  of  a  Siege ;  III ,  School  of 
the  Sap ;  IV.,  Military  Mining ;  V.,  Construction  of  Batteries.  By 
Major  J.  C.  DUANE,  Corps  of  Engineers,  U.  S.  Army.  1  vol.  12mo, 
cloth.  $2.50. 

"I  have  carefully  examined  Capt.  J.  C.  Duane's  'Manual  for  Engineer  Troops,"1  and  do 
not  hesitate  to  pronounce  it  the  very  best  work  on  the  subject  of  which  it  treats. 

"H.  W.  HALLECK,  Major- General,  U.  S.  A:"1 

Diifoiir's  Principles  of  Strategy  and  Grand  Tactics. 

Translated  from  the  French  of  General  G.  H.  Dufour.  By  WILLIAM  P. 
CRAIGHILL,  Capt.  of  Engineers,  U.  S.  Army,  and  Assistant  Professor  of 
Engineering,  U.  S.  Military  Academy,  West  Point.  From  the  last 
French  Edition.  Illustrated.  In  one  volume  12mo.  cloth.  $3.00. 

"  In  all  military  matters  General  Dufour  is  recognized  as  one  of  the  first  authorities  fa 
Europe,  and  consequently  the  translation  of  this  very  valuable  work  is  a  most  acceptable 
addition  to  our  military  libraries." — London  Naval  and  Military  Gazette. 

11 


D.  Van  Nostrandl  s  Publications. 


Iii§tructions  for  Field  Artillery.  Prepared  by  a  Board  of 
Artillery  Officers.  1  voL  12mo,  illustrated  by  122  pages  of  Engravings. 
Cloth.  $3.00. 

"WAR  DEPARTMENT,  WASHINGTON,  March,  1, 1868. 

"This  system  of  Instruction  for  Field  Artillery,  prepared  under  direction  of  the  War 
Department,  having  been  approved  by  the  President,  is  adopted  for  the  instruction  of 
troops  when  acting  as  field  artillery. 

"  Accordingly,  instruction  in  the  same  will  be  given  after  the  method  pointed  out 
therein ;  and  all  additions  to  or  departures  from  the  exercise  and  manoeuvres  laid  dowu 
in  the  system,  are  positively  forbidden. 

"EDWIN  M.  STANTON,  Secretary  of  War." 

Anderson's  Evolutions  of  Field  Batteries  of  Artil- 
lery. Translated  from  the  French,  and  arranged  for  the  Army  and 
Militia  of  the  United  States.  By  Gen.  ROBERT  ANDERSON,  U.  S.  A. 
Published  by  order  of  the  War  Department.  1  vol.,  cloth,  32  plates.  $1. 

"  WAR  DEPARTMENT,  Nov.  2d,  1859. 

"The  system  of  'Evolutions  of  Field  Batteries,'  translated  from  the  French,  and  ar- 
ranged for  the  service  of  the  United  States,  by  Major  Kobert  Anderson,  of  the  1st  Begi- 
ment  of  Artillery,  having  been  approved  by  the  President,  is  published  for  the  informa- 
tion and  government  of  the  army. 

"All  Evolutions  of  Field  Batteries  not  embraced  in  this  system  are  prohibited,  and 
those  herein  prescribed  will  be  strictly  observed.  J.  B.  FLOYD, 

"•Secretary  of  War." 

Mendel  1's  Treatise  on  Military  Surveying.  Theoretical 
and  Practical,  including  a  Description  of  Surveying  Instruments.  By 
G.  H.  MENDELL,  Captain  of  Engineers.  1  vol.  8vo,  with  numerous  illus- 
trations, cloth.  $2.50. 

"The  author  is  a  Captain  of  Engineers,  and  has  for  his  chief  authorities  Salneuve,  Lalo- 
bre,  and  Simms.  He  has  presented  the  subject  in  a  simple  form,  and  has  liberally  illus- 
trated it  with  diagrams,  that  it  may  be  readily  comprehended  by  every  one  who  is  liable 
to  be  called  upon  to  furnish  a  military  sketch  of  a  portion  of  country." — N&io  York 
Evening  Post. 

Viele's  Hand-book.  Hand-Book  for  Active  Service,  containing 
Practical  Instructions  in  Campaign  Duties.  For  the  use  of  Volunteers. 
By  Brigadier-General  Egbert  L.  Viele,  U.  S.  A.  12mo,  cloth.  $1. 

M  It  is  a  thorough  treatise,  copiously  illustrated,  and  embraces  a  complete  drill  by  com- 
pany, regiment,  &c.  It  also  embraces  instructions  in  regard  to  the  camp,  fortifications, 
rations,  and  mode  of  cooking  them,  and  has  a  manual  for  light  and  heavy  artillery." — 
New  Haven  Palladium. 

A  Treatise  on  the  Camp  and  March.  "With  which  is 
connected  the  Construction  of  Field- Works  and  Military  Bridges ;  with  an 
Appendix  of  Artillery  Ranges,  &c.  For  the  use  of  Volunteers  and  Militia 
in  the  United  States.  By  Captain  Henry  D.  Grafton,  U.  S.  A.  1  voL 

12mo.  cloth.  75  cents. 

12 


D.  Van  Nostrand?  s  Publications. 

It    i 
The  Automaton  Regiment;    or,   Infantry   Soldier's 
Practical   Instructor. — For  all  Regimental  Movements   in  the 
Field,    By  G-.  Douglas  Brewerton,  U.  S.  Army.    Neatly  put  up  in  boxes, 
price  $1 ;    when  sent  by  mail,  $1.40. 

The  "Automaton  Regiment"  is  a  simple  combination  of  blocks  and  counters,  so 
arranged  and  designated,  by  a  carefully  considered  contrast  of  colors,  that  it  supplies  the 
student  with  a  perfect  miniature  regiment,  in  which  the  position  in  the  battalion  of  each 
company,  and  of  every  officer  and  man  in  each  division,  company,  platoon,  and  section, 
is  clearly  indicated.  It  supplies  the  studious  soldier  with  the  means  whereby  he  can 
consult  his  "tactics,"  and  at  the  same  time  join  practice  to  theory  by  manoeuvring  a 
mimic  regiment. 

I  hereby  certify  that  I  have  examined  the  "  Automaton  Regiment,"  invented  by  GL 
Douglas  Brewerton,  late  of  the  United  States  Regular  Army,  and  now  serving  as  a  Vol- 
unteer Aide  upon  my  military  staff,  and  believe  that  his  invention  will  prove  a  useful 
and  valuable  assistant  to  every  student  of  military  tactics.  I  take  pleasure  in  recom- 
mending it  accordingly.  B.  SAXTON, 

Brigadier-  General  Volunteers. 

The  Automaton  Company;  or,  Infantry  Soldier's 
Practical  Instructor. — For  all  Company  Movements  in  the  Field. 
By  G-.  Douglas  Brewerton,  U.  S.  A.  Price  in  boxes,  $1.25 ;  when  sent 
by  mail,  $1.95. 

The  Automaton  Battery;  or,  Artillerist's  Practical 
Instructor. — For  all  Mounted  Artillery  Manoeuvres  in  the  Field. 
By  G-.  Douglas  Brewerton,  U.  S.  A.  Price  in  boxes,  $1 ;  when  sent  by 
mail,  $1.40. 

These  productions  are  of  a  similar  character,  and  cannot  fail  to  be  of  great  value  to  the 
military  student.  They  are  object  lessons,  that  will  teach  more  in  an  hour  than  mere 
rerbal  instruction  could  in  a  month.  They  cannot  be  too  highly  commended  to  both 
officers  and  men. — Commercial  Advertiser. 

Monroe's  Company  and  Skirmish  Drill.— The  Company 
Drill  of  the  Infantry  of  the  Line,  together  with  the  Skirmish  Drill' of  the 
Company  and  Battalion,  after  the  method  of  G-eneral  Le  Louterel.  Bayonet 
Fencing ;  with  a  Supplement  on  the  Handling  and  Service  of  Light  In- 
fantry. By  J.  Monroe,  Colonel  22d  Regiment,  N.  G-.,  N.  Y.  S.  M.,  for- 
merly  Captain  U.  S.  Infantry.  1  vol.  32mo,  cloth.  75  cents. 

This  is  a  most  valuable  and  timely  little  manial.  It  should  be  in  the  hands  of  every 
new  recruit. — Chicago  Tribune. 

School  of  the  Guides. — Designed  for  the  use  of  the  Militia  of  the 
United  States.  By  Colonel  Eugene  Le  Gal,  55th  Regiment,  N.  Y.  S. 
M.  cloth,  60  cents. 

"  This  excellent  compilation  condenses  into  a  compass  of  less  than  sixty  pages  all  the 
Instruction  necessary  for  the  guides,  and  the  information  being  disconnected  with  other 
matters,  is  more  readily  referred  to  and  more  easily  acquired." — Louisville  Journal. 

L- i? 


D.  Van  Nostrarid?  s  Publications. 


Craighill's  Army  Officer's  Pocket  Companion.  Princi- 
pally designed  for  Staff  Officers  in  the  Field.  Partly  translated  from  the 
French  of  M.  DE  ROUVRE,  Lieutenant- Colonel  of  the  French  Staff  Corps, 
with  Additions  from  standard  American,  French,  and  English  Authorities. 
By  T\rM.  P.  CRAIGHILL,  First  Lieutenant  U.  S.  Corps  of  Engineers,  Assist. 
Prof,  of  Engineering  at  the  U.  S.  Military  Academy,  West  Point.  1  vol. 
18mo,  full  roan.  $2.00. 

"  I  have  carefully  examined  Capt.  Craighill's  Pocket  Companion.  I  find  it  one  of  the 
very  best  works  of  the  kind  I  have  ever  seen.  Any  Army  or  Volunteer  officer  who  will 
make  himself  acquainted  with  the  contents  of  this  little  book,  will  seldom  be  ignorant 
of  his  duties  in  camp  or  field.  II.  W.  HALLECK,  Major- General  U.  S.  A." 

Hunter's  Manual  for  Quartermasters  and  Commis- 
saries. Containing  Instructions  in  the  Preparation  of  Touchers,  Ab- 
stracts, Returns,  etc.,  embracing  all  the  recent  changes  in  the  Army 
Eegulations,  together  with  instructions  respecting  Taxation  of  Salaries, 
etc.  By  Captain  R.  F.  HUNTER,  late  of  the  U.  S.  Army.  12mo,  cloth, 
$1.25.  Flexible  morocco,  $1.50. 

"This  is  the  only  work  of  the  kind  extant.  It  is  based  on  the  latest  regulations  of  the 
War  Department,  and  will  be  regarded  as  authority  by  those  officers  for  whose  use  it  is 
designed."— Saturday  Evening  Gazette. 

Ordronaux's  manual  of  Instructions  for  Military 
Surgeons,  in  the  Examination  of  Recruits  and  Discharge  of  Sol- 
diers. With  an  Appendix,  containing  the  Official  Regulations  of  the 
Provost-Marshal  General's  Bureau,  and  those  for  the  formation  of  the  In- 
valid Corps,  etc.,  etc.  Prepared  at  the  request  of  the  United  States 
Sanitary  Commission.  By  JOHN  ORDRONAUX,  M.  D.,  Professor  of  Medical 
Jurisprudence  in  Columbia  College,  New  York.  12mo.  Half  morocco. 
$1.50. 

"  In  a  condensed  form,  it  is  an  admirable  treatise  on  the  important  subjects  of  which  it 
treats.  The  author  has  aimed  to  be  brief  without  being  obscure,  to  omit  nothing  of  real 
importance,  and  to  draw  his  materials  from  the  best  sources.  He  treats  of  the  physical 
disabilities  which  have  relation  to  the  military  service,  and  of  these  alone.  Medical  Ex- 
aminers are  instructed  in  their  duties,  and  the  method  of  discovering  feigned,  artificially 
produced,  and  concealed  diseases  is  pointed  out  The  book  will  prove  valuable  to  all 
who  are  concerned  in  the  manipulation  of  recruits  or  conscripts.  An  Appendix  contains 
official  regulations  and  instructions  relative  to  the  Provost-Marshal's  office,  the  Invalid 
j  Corps,  etc." — Commercial  Advertiser. 

\  Ordronaux's  Hints  on  the  Preservation  of  Health  in 
Armies.  For  the  use  of  Volunteer  Officers  and  Soldiers.  By  JOHN 
ORDRONAUX,  M.  D.  New  edition,  18mo,  cloth.  75  cents. 

14 


D.  Van  No  sir  and?  s  Publications. 


Thomas's  Rifled  Ordnance.  A  Practical  Treatise  on  the  Ap- 
plication of  the  Principle  of  the  Rifle  to  Guns  and  Mortars  of  every 
Calibre.  To  which  is  added,  a  new  theory  of  the  initial  action  and  force 
of  Fired  Gunpowder.  By  LYNALL  THOMAS,  F.  R.  S.  L.  Fifth  edition, 
revised.  One  volume,  octavo,  illustrated,  cloth-  $2.00. 

"At  a  time  when  the  manufacture  of  guns  engrosses  the  attention  of  thousands  on 
thousands,  any  practical  treatise  which  may  suggest  desirable  alterations  or  innovations 
is  of  importance,  and  deserves  that  attention  we  doubt  not  will  be  extended  to  the  pres- 
ent volume." — Boston  Evening  Gazette. 


Wilcox's  Rifles  and  Rifle  Practice.  An  Elementary  Trea- 
tise on  the  Theory  of  Rifle  Firing ;  explaining  the  causes  of  Inaccuracy 
of  Fire  and  the  manner  of  correcting  it ;  with  descriptions  of  the  Infan- 
try Rifles  of  Europe  and  the  United  States,  their  Balls  and  Cartridges. 
By  Captain  C.  M.  WILCOX,  U.  S.  A.  New  edition,  with  engravings  and 
cuts.  Green  cloth.  $2.00. 

!  "  The  book  will  be  found  intensely  interesting  to  all  who  are  watching  the  changes  in 

the  art  of  war  arising  from  the  introduction  of  the  new  rifled  arms.    We  recommend  to 
our  readers  to  buy  the  book." — Military  Gazette. 


Lcmly's  Maxims  and  Instructions  on  the  Art  of  War. 

Maxims,  Advice,  and  Instructions  on  the  Art  of  War ;  or,  A  Practical 
Military  Guide  for  the  use  of  Soldiers  of  all  Arms  and  of  all  Countries. 
Translated  from  the  French  by  Captain  LENDY,  Director  of  the  Practical 
Military  College,  late  of  the  French  Staff,  etc.,  etc.  1  vol.  18mo,  cloth. 
75  cents. 

"This  book  treats  generally  of  the  art  of  war  and  the  conduct  of  campaigns,  and  with- 
out going  into  all  the  details  of  a  soldier's  business,  aims  to  explain  the  principles  on 
which  an  army  may  be  well  established  in  camp,  or  successfully  led  and  mano3uvred  on 
the  field." — Providence  Journal. 


Andrews's  Hints  to  Company  Officers  oil  their  Mili- 
tary Duties.  By  Captain  C.  C.  ANDREWS,  Third  Regiment  Minne- 
sota Volunteers.  1  vol.  18rno,  cloth.  60  cents. 

"This  is  a  hand-book  of  good  practical  advice,  which  officers  of  all  ranks  may  study 
with  advantage.'' — Philadelphia  Press. 


Ileth's  System  of  Target  Practice.— For  the  use  of  Troops 
when  armed  with  the  Musket,  Rifle-Musket,  Rifle,  or  Carbine.  Prepared 
principally  from  the  French,  by  Captain  Henry  Heth,  10th  Infantry,  TJ  S. 
A.  18mo,  cloth.  75  cents. 

15 


D.   Van  No  sir  and}  s  Publications. 

Miiiifie's  Mechanical  Drawing.  A  Text-Book  of  Geometrical 
Drawing,  for  the  Use  of  Mechanics  and  Schools,  in  which  the  Definitions 
and  Rules  of  Geometry  are  familiarly  explained;  the  Practical  Prob- 
lems are  arranged  from  the  most  simple  to  the  more  complex,  and  in  their 
description  technicalities  are  avoided  as  much  as  possible.  With  Illus- 
trations for  Drawing  Plans,  Sections,  and  Elevations  of  Buildings  and 
Machinery;  an  introduction  to  Isometrical  Drawing,  and  an  Essay  on 
Linear  Perspective  and  Shadows.  Illustrated  with  over  200  Diagrams 
engraved  on  Steel.  By  WILLIAM  MINIFIE,  Architect.  Seventh  edition. 
With  an  Appendix  on  the  Theory  and  Application  of  Colors.  1  vol.  8vo, 
cloth.  $4.00. 

"  It  is  the  best  work  on  Drawing  that  we  have  ever  seen,  and  is  especially  a  text-book 
of  Geometrical  Drawing  for  the  use  of  Mechanics  and  Schools.  No  young  Mechanic,  such 
as  a  Machinist,  Engineer,  Cabinet-Maker,  Millwright,  or  Carpenter,  should  be  without  it.1' 
— Scientific  American. 

"One  of  the  most  comprehensive  works  of  the  kind  ever  published,  and  cannot  but 
possess  great  value  to  builders.  The  style  is  at  once  elegant  and  substantial." — Penn- 
sylvania Inquirer. 

"We  think  this  the  best  work  on  this  subject,  which  is  saying  very  much  ;  as  much 
attention  has  been  given  to  the  science  of  Drawing.  There  is  nothing  in  the  range  of 
drawing  that  cannot  be  found  in  this  book,  and  which  is  not  well  explained."—  Ohio 
Teacfter. 

u  Whatever  is  said  is  rendered  perfectly  intelligible  by  remarkably  well  executed  dia- 
grams on  steel,  leaving  nothing  for  mere  vague  supposition ;  and  the  addition  of  an 
introduction  to  isometrical  drawing,  linear  perspective,  and  the  projection  of  shadows, 
winding  up  with  a  useful  index  to  technical  terms." — Glasgow  Mechanic's  Journal. 


Minified  Geometrical  Drawing.  Abridged  from  the  Octavo 
edition,  for  the  use  of  Schools.  Illustrated  with  48  Steel  Plates.  Fifth 
edition.  1  vol.  12mo,  half  roan.  $1.50. 

Peirce'§  System  of  Analytic  Mechanics.  Physical  and  Ce- 
lesrial  Mechanics,  by  BENJAMIN  PEIRCE,  Perkins  Professor  of  Astronomy 
and  Mathematics  in  Harvard  University,  and  Consulting  Astronomer  of 
the  American  Ephemeris  and  Nautical  Almanac.  Developed  in  four  sys- 
tems of  Analytic  Mechanics,  Celestial  Mechanics,  Potential  Physics,  and 
Analytic  Morphology.  1  vol  4to(  cloth.  $10.00. 

Dictionary  of  Weights  and  Measures.  Universal  Dictionary 
of  Weights  and  Measures,  Ancient  and  Modern,  reduced  to  tie  standards 
of  the  United  States  of  America.  By  J.  H.  ALEXANDER.  New  edition, 
enlarged.  1  vol.  8vo,  cloth.  $2 .50 

16 


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